I$mmmmmmmmmmimmmmmimmimu iiiiiiiiiiiiiiiiMmiiiitiiymnMwiwwiiiwtimi^wwB^^ 


AN  INTRODUCTORY 

ARITHMETIC 


wiwiiaiiwiimiin; 


SENSENIG^ARDERSOH 


jBft^Buiiirr  ^^fim 


iso 


DEI 


No.    ^ 


H 


5[|)c  Sinocnig-^nberson  Scrica  of  ^IritlimEticB 


mTRODUCTORY  ARITHMETIC 

BY 
DAYID  M.  SE^N^SEKIG,  M.S. 


KOBEET  F.  ANDERSOJS^,  A.M. 

INSTRUCTORS  IN  MATHEMATICS,  STATE  NORMAL  SCHOOL, 
WEST  CHESTER,  PENNSYLVANIA 


SILVER,    BURDETT    AKD    COMPAlSrY 

NEW  YORK  BOSTON  CHICAGO 


aAio3 

S4- 


Ete  Srngenifi'^ntiergon  Series  of  arithmetics 

By  DAVID  M.  SENSENIG,  M.S.,  and  ROBERT  F. 
ANDERSON,  A.M.,  Instructors  in  Mathematics, 
State  Normal  School,  "West  Chester,  Pennsylvania. 


AN  INTRODUCTORY  ARITHMETIC.  A  first 
book  in  arithmetic,  inductive  in  method  and  carefully 
graded,  developing  the  subjects  by  natural  steps.  It 
gives  the  pupil  an  intelligent  understanding  of  processes 
and  abundant  practice  in  operations.     262  pp. 

ESSENTIALS  OF  ARITHMETIC.  A  compre- 
hensive and  practical  book  for  grammar  grades,  giving 
thorough  instruction  and  drill  in  fundamental  processes 
and  much  information  that  is  useful  in  business  transac- 
tions. The  treatment  of  mensuration  leads  up  to  algebra 
and  geometry.     343  pp.    60  cents. 

THE     NEW     COMPLETE     ARITHMETIC    for 

High  Schools,  Academies,  and  Normal  Schools.  A  prac- 
tical text-book  in  which,  besides  the  usual  topics,  special 
attention  is  given  to  business  papers,  with  photographic 
reproductions  of  business  forms,  to  property  and  life 
assurance,  and  mensuration.    437  pp.    90  cents. 


SILVER,  BURDETT  AND  COMPANY 

New  York  Boston  Chicago 


eOUCATION  DCTf  I 


Copyrighi,  1903, 
By  Silver,  Burdett  and  Company 


<UL 


PREFACE 


Iij  this  volume  the  authors  assume  that  the  pupil  can 
read  easy  sentences  and  that  he  has  had  at  least  one  year 
of  number  work. 

The  aim  of  Chapter  I.  is  to  develop  easily  and  naturally 
the  idea  of  number  without  making  use  of  elaborate  pic- 
tures; to  provide  numerous  exercises,  both  oral  and  writ- 
ten, by  means  of  which  the  pupil  may  become  proficient  in 
forming,  writing,  and  reading  numbers,  and  in  the  funda- 
mental operations;  to  furnish  him  with  such  suggestions 
and  solutions  as  will  make  each  step  in  his  progress  in- 
telligible to  him. 

Other  important  features  of  the  book  are: 

1.  The  inductive  methods  employed  in  leading  the  pupil 
by  logical  questions  to  the  fundamental  conceptions  of 
every  subject. 

2.  The  use  of  simple  diagrams,  easily  constructed  by  the 
pupil  himself,  for  purposes  of  illustration. 

3.  Carefully  graded  concrete  problems,  involving  only 
such  terms  as  the  pupil  is  supposed  to  be  familiar  with. 
lu  other  words,  the  authors  have  endeavored  to  keep  within 
the  field  of  the  pupil's  experience. 

54  J  543 


iv  PREFACE. 


4.  The  separation  of  the  process  of  finding  one  of  the 
equal  parts  of  a  number.  Division,  and  of  finding  how 
many  times  one  number  contains  another  as  a  unit  of 
measure,  Mensuration.  Division  as  usually  treated  in- 
volves two  cases  so  obviously  different  in  their  nature  that 
they  ought  to  be  considered  separate  processes  under  ap- 
propriate names.  The  process  of  separating  a  number  into 
a  number  of  equal  parts  to  determine  one  of  these  parts  is 
properly  named  Division.  The  process  of  determining 
how  many  times  one  number  contains  another  as  a  unit  of 
measure,  which  is  generally  called  division,  is  primarily 
finding  the  relation  which  one  number  bears  to  another 
taken  as  a  unit  of  measure;  this  is  simply  measuring  one 
number  by  another,  and  is  properly  named  Mensuration. 

5.  A  simple  treatment  of  Percentage,  Interest,  and  Busi- 
ness Forms,  introduced  to  meet  the  demands  of  such  pupils 
as  are  compelled  to  leave  school  at  an  early  age. 

David  M.  Sensenig. 
RoBEKT  F.  Anderson. 


CONTENTS. 


CHAPTER  I. 

Page 
Whole  Numbers  and   Fractional  Parts   of  Whole  Num- 
bers       1-143 

Forming,  Writing,  and  Reading  Numbers  to  10  ;  Addition 

and  Subtraction 1 

Forming,  Writing,  and  Reading  Numbers  from  10  to  20  ; 

Addition  and  Subtraction 13 

Forming,  Writing,  and  Reading  Numbers  from  20  to  100  ; 

Addition  and  Subtraction 28 

Forming,    Writing,   and   Reading   Numbers  from   100  to 

1000  ;  Addition  and  Subtraction 36 

Multiplication,  Division  and  Mensuration 47 

Forming,  Writing,  and  Reading  Numbers  from  1000  to  10000  78 

Roman  Notation       79 

United  States  Money 81 

Making  Change 83 

Addition  and  Subtraction 84 

Multiplication,  Division  and  Mensuration 89 

Forming,   Writing  and  Reading  Numbers  above  10000  ; 

Fundamental  Operations 132 

Factors 141 

Divisors 143 


CHAPTER  II. 

Fractions 144-176 

Introductory  Problems  and  Definitions 144 

Reduction 147 

Multiplication  and  Division 151 

V 


vi  CONTENTS. 


Fractions  —  continued.  Paob 

Reduction 154 

Addition  and  Subtraction 156 

Multiplication 161 

Division  and  Mensuration 168 

Complex  Fractions 176 

CHAPTER    III. 

Decimals 177-201 

Introductory  Problems  and  Definitions 177 

Reading  and  Writing  Decimals 178 

Reduction 182 

Addition  and  Subtraction 185 

Multiplication,  Division  and  Mensuration 186 

CHAPTER  IV 

Denominate  Amounts     .         202-240 

United  States  Money 202 

Bills  and  Accounts 206 

Measures  of  Time 209 

Measures  of  Capacity 211 

Measures  of  Weight 213 

Measures  of  Length 216 

Measures  of  Surface 221 

Cubic  Measure 227 

Rectangles 230 

Carpeting 232 

Lumber  Measure 233 

Rectangular  Solids 235 

Compound  Denominate  Amounts 237 

CHAPTER  V. 

Percentage 241-252 

Interest 247 

Business  Papers 249 

Miscellaneous  Problems 252 


An  Introductory  Arithmetic. 

^ 

CHAPTER   I. 

WHOLE   NUMBERS  AND  FRACTIONAL  PARTS 
OF  WHOLE   NUMBERS. 

Forming,  Writing,  and    Reading  Numbers  to   10; 
Addition  and  Subtraction. 

1.  Oral  Exercise. 

How  many  dots  are  there  in  this  square  ?  *  I 

The  figure  i  stands  for  one. 

How  many  are  1  and  1  ?  ** 

The  figure  2  stands  for  two. 


How  many  are  2  and  1  ?    1  and  2  ? 
The  figure  3  stands  for  three. 

How  many  are  3  and  1  ?    1  and  3  ? 
How  many  are  2  and  2  ? 
The  figure  4  stands  for  four. 


•  •• 


.a*. : WH;fc>i'5J.NtJ>ifiEHa  and  fractional  parts. 


IIow  many  are  4  and  1  ?     1  and  4  ? 
How  many  are  3  and  2  ?    2  and  3  ? 
The  figure  5  stands  for  five. 

How  many  are  5  and  1  ?    1  and  5  ? 
How  many  are  4  and  2  ?    2  and  4  ? 
How  many  are  3  and  3  ? 
The  figure  6  stands  for  six. 

How  many  are  6  and  1  ?    1  and  G  ? 

How  many  are  5  and  2  ?    2  and  5  ? 

How  many  are  4  and  3  ?    3  and  4  ? 

The  figure  7  stands  for  seven. 

How  many  are  7  and  1  ?    1  and  7  ? 
How  many  are  6  and  2  ?    2  and  6  ? 
How  many  are  5  and  3  ?    3  and  5  ? 
How  many  are  4  and  4  ? 
The  figure  8  stands  for  eight. 


• 
•  ••• 

•  • 
•  •  • 

• 

•  • 
•  ••• 

1  ••• 

• 

•• 

••• 
•••• 

• 

•  • 

•  •  • 

•  •  •  4 

•  •  •  • 

NUMBERS  T0^>0: 


How  many  are  8  and  1  ?  1  and  8  ? 
How  many  are  7  and  2?  2  and  7  ? 
How  many  are  6  and  3  ?  3  and  6  ? 
How  many  are  5  and  4  ?  4  and  5  ? 
The  figure  g  stands  for  nine. 

2  pins  and  2  pins  are  how  many  pins  ? 


• 

•  • 

•  •  • 

•  •  • 
•  •  •  • 

• 
• 

nil 


How  many  marbles  are  3  marbles  and  2  marbles  ? 

2  pears  and  3  pears  are  how  many  pears  ?    9^9     9^ 

4  rings  and  2  rings  are  how  many  rings  ? 

How  many  pies  are  2  pies  and  4  pies  ?        OOOO     OO 

How  many  cards  are  3  cards  and  3  cards  ?  nnri     ["100 


5  balls  and  2  balls  are  how  many  balls  ? 
2  oranges  and  5  oranges  are  how 
many  oranges  ? 


How  many  pencils  are  4  pencils  and  3  pencils  ? 
3  girls  and  4  girls  are  how  many  girls  ? 

6  crosses  and  2  crosses  are  how  many  crosses  ? 
2  poles  and  6  poles  are  how  many 


poles 


XXXXXX    XX 


'4'*.:'i^Hi)Li:''i^im.i5?itS.AND  fractional  parts. 

5  books  and  3  books  are  how  many  books  ? 

How  many  letters  are  3  letters  and  rirnririri     nnn 
o  letters  i 

////  //// 


How  many  fingers  are  4  fingers  and  4 
fingers  ? 


7  stamps  and  2  stamps  are  how  many  stamps  ? 

How  many  beans  are  2  beans  and  7    /^^^y^^^^     ^^ 
beans?  0000000    00 

6  apples  and  3  apples  are  how  many 

apples  P  \\\\\\\\\ 

How  many  tops  are  3  tops  and  6  tops  ? 
How  many  logs  are  5  logs  and  4  logs  ? 
How  many  beads  are  4  beads  and  5  beads  ?       |nJ  I 

2.  The  sign  +  stands  for  the  word  and. 
The  sign  =  stands  for  are  or  is. 

Thus,  2  and  1  are  3  may  be  written  2  -\-  1  =  S. 

3.  Oral  Exercise. 

Read  : 

1+1=2  3+1=4  4+1=5 
1+2=3  1+3=4  1+4+5 
2+2=4      3+2=5      5+1=6 

4.  Exercise. 

Copy,  writing  the  proper  figures  in  place  of  the  dots: 


ADDITION  AND  SUBTRACTION. 


1  +  1  =  • 

4  +  2=  . 

2  +  6=  . 

2  +  1=  • 

2  +  4=  « 

5  +  3  =  . 

1  +  2  = 

3  +  3  = 

3  +  5  = 

3  +  1=  . 

6  +  1  = 

4  +  4  = 

1  +  3=: 

1  +  6  = 

8  +  1=  . 

2  +  2  = 

5  +  2  =  • 

1  +  8  = 

4  +  1  =  . 

2  +  5  =  . 

7  +  2=  . 

1  +  4== 

4  +  3  =  . 

2  +  7=  « 

3  +  2  = 

3  +  4  = 

6  +  3  = 

2  +  3  =  . 

7  +  1=  . 

3  +  6  = 

5  +  1  = 

1  +  7  = 

5  +4=  . 

1  +  5  = 

6  +  2=  . 

4  +  5  =  . 

5.  Oral  Exercise. 

Eead  Exercise  4,  supplying  the  missing  numbers. 

6.  Exercise. 

Copy,  writing  in  each  case  the  proper  figure  in  place  of 
the  dash  : 

2  men  -\ men  =  4  men. 

3  women  H women  =  5  women. 

2  boys  H boys  =  5  boys. 

4  girls  H girls  =  6  girls. 

2  horses  -{ horses  =  6  horses. 

3  cows  H cows  =  6  cows. 

5  mules  -\ mules  =  7  mules. 

2  goats  -\ goats  =  7  goats. 

4  kittens  H kittens  =  7  kittens. 


ooooo 

^Ei^^  ^b  ^^  ht^  ^b 


6      WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 

'>  hens  H hens  =  7  hens. 

6  ducks  -] ducks  =  8  ducks. 

2  geese  -\ geese  =  8  geese. 

5  turkeys  -I turkeys  =  8  turkeys.  ####### 


3  crows  +  —  crows  =  8  crows. 

4  wrens  H wrens  =  8  wrens. 

7  robins  H — -  robins  =  9  robins. 

2  doves  H doves  =  9  doves, 

6  jays  +  —  jays  =  9  jays.  #1 

3  sparrows  -I sparrows  =  9  sparrows. 

5  parrots  H parrots  =  9  parrots. 

4  larks  H larks  =  9  larks. 


7.  Exercise. 

Copy,  writing  the  proper  figure  in  place  of  the  dots 


1  +  • 

=  2 

4  + 

.   =  G 

2  + 

.  =8 

2+  • 

=  3 

2  +  • 

=  6 

5  + 

=  8 

1  +  • 

=  3 

3  +  . 

=  G 

3  + 

.   =8 

3  +  . 

=  4 

c  +  . 

=  7 

4  +  . 

=  8 

1  +  • 

=  4 

1  +  • 

=  7 

8  + 

'   =  9 

2+  . 

=  4 

5  +  . 

=  7 

1  + 

=  9 

4  +  . 

=^5 

2  + 

•  =  7 

7  + 

.   =9 

1  +  ' 

=  5 

4  + 

•  =  7 

2  + 

.  =9 

3  + 

.   =5 

3  + 

.  =  7 

G  + 

.   =9 

2  + 

.   =5 

7  + 

.  =8 

3  + 

.   =9 

5  + 

.  =G 

1  + 

.  =8 

5  + 

.   =9 

1  + 

►  =  6 

C  + 

.   =8 

4  + 

.  =9 

ADDITION  AND  SUBTRACTION. 


8.  Oral  Exercise. 

I  Read  Exercise  7,  supplying  the  missing  numbers. 

9.  Exercise. 

Copy,  writing  in  each  case  the  proper  figure  in  place  of 
the  dash  : 

—  birds  +  2  birds  =  4  birds. 

—  mice  +  2  mice  =  5  mice. 

—  rats  +  3  rats  =  5  rats. 

—  bugs  +  2  bugs  —  6  bugs. 

—  flies  +  4  flies  =  6  flies. 

—  fishes  +  3  fishes  =  6  fishes. 

—  lambs  +  2  lambs  =  7  lambs. 

—  trees  +  5  trees  =  7  trees. 

—  shrubs  +  3  shrubs  =  7  shrubs. 

—  pinks  +  4  pinks  =  7  pinks. 

—  daisies  +  2  daisies  =  8  daisies. 

—  buds  4-  6  buds  =  8  buds. 

—  leaves  +  3  leaves  =  8  leaves. 

—  roses  4-  5  roses  =  8  roses. 

—  violets  +  4  violets  =  8  violets. 

—  cars  +  2  cars  =  9  cars. 

—  carts  +  7  carts  —  9  carts. 

—  ships  +  3  ships  =  9  ships.  ( 

—  boats  +  G  boats  =  9  boats. 

—  bricks  4-  4  bricks  =  9  bricks. 

—  stones  +  5  stones  =  9  stones. 


ooooo 

iXi  aSU  ^b  Ai  wT^  i^r^ 

^r  \r  ^r  tc  t?  <p 

/////// 


8      WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 


10.  Exercise. 

Copy,  writing  the  proper  figures  in  place  of  the  dots 


.  +  1  =  2 

.  +  2  =  6 

.  +6  =  8 

.  +  1  =  3 

.  +4  =  6 

.  +  3  =  8 

•  +2  =  3 

.+3  =  6 

.+5  =  8 

.  +1  =  4 

.  +  1  =  7 

.  +4  =  8 

.  +3  =  4 

.  +  6  =  7 

.+1  =  0 

.  +  2  =  4 

►  +  2  =  7 

.  +  8  =  9 

.  +  1  =  5 

►  +5  =  7 

.+2  =  9 

.  +  4  =  5 

►  +3  =  7 

.+7  =  9 

.  +  2  =  5 

.  +4  =  7 

-r  3  =  9 

.  +  3  =  5 

.  +  1  =  8 

+  6  =  9 

.+1  =  6 

+  7  =  8 

+  4  =  9 

.+5  =  6 

+  2  =  8 

+  5  =  9 

11.  Oral  Exercise. 

Bead  Exercise  10,  supplying  the  missing  numbers. 

12.  Oral  Exercise. 

Read,  supplying  the  missing  numbers  : 

4  nails  less  2  nails  are  —  nails.  I  I  I  I 

5  grains  of  corn  less  2  grains  of  corn  are  —  grains  of  corn. 

5  pods  less  3  pods  are  —  pods.  T  T  T  T  T 

6  spoons  less  2  spoons  are  —  spoons. 

6  forks  less  4  forks  are  —  forks.  OOOOOO 

6  cups  less  3  cups  are  —  cups. 

7  knives  less  2  knives  are  —  knives. 

7  spoons  less  5  spoons  are  —  spoons.    ^A  A  A  A  A  A 
7  fishes  less  3  fishes  are  —  fishes. 


ADDITION  AND  SUBTRACTION. 


9 


7  eels  less  4  eels  are  —  eels. 

8  cents  less  2  cents  are  —  cents. 

8  dollars  less  6  dollars  are  —  dollars. 

8  cars  less  3  cars  are  —  cars.  XXXXXXXX 

8  carts  less  5  carts  are  —  carts. 

9  cards  less  2  cards  are  —  cards. 

9  spools  less  7  spools  are  —  spools.  #•####### 
9  matches  less  3  matches  are  —  matches. 
9  brooms  less  6  brooms  are  —  brooms. 
9  turkeys  less  4  turkeys  are  —  turkeys. 
9  chairs  less  5  chairs  are  —  chairs. 

13.  The  sign  —  stands  for  the  word  less. 
Tims,  6  less  4  are  2  may  be  written  6  —  Ji.  =  2. 

14.  Oral  Exercise. 
Eead : 

2-1  =  1  4-1:^3  5-1  =  4 

3-1  =  2  4-2  =  2  5-2  =  3 

3-2  =  1  4-3  =  1  5-3  =  2 


15.  Exercise. 

Copy,  writing  the  proper  figures  in  place  of  the  dots  : 


2-1  = 
3  -  1  = 
3-2  = 
4-1  = 
4-3  = 
4-2  = 
5  -  1  = 


5-4  = 

5-2  = 

5  -  3  = 

6  -  1  = 
6  -  5  = 
G  -  2  = 
6-4  = 


G  -  3  = 
7-1  = 

7-6  = 

7-2  = 
7  -  5  = 
7-3  = 
7-4  = 


10  WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 


8-  1  =  . 

8-5  = 

9-7=  . 

8-7=  o 

8  -4=  . 

9-3  =  . 

8-2=  o 

9  -  1  =  . 

9- 6=  . 

8-  G  =  . 

9-  8  = 

9  -4=  . 

8-3=  • 

9-2=  . 

9-5=  . 

16.  Oral  Exercise. 

Read  Exercise  15,  supplying  the  missing  numbers. 

17.  Oral  Exercise. 

How  many  stars  are  2  stars  and  2  stars  ? 

How  many  more  than  2  dots  are  4  dots  ? 

How  many  rings  are  3  rings  and  2  rings  ? 

How  many  squares  must  be  put  with  3  squares  to  make  5 

squares  ? 
How  many  more  than  2  blocks  are  5  blocks  ? 

How  many  strokes  are  here  ?  ////// 

If  you  take  away  4  strokes,  how  many  will  be  left  ? 
How  many  more  than  2  strokes  are  here  .^ 
How  many  more  than  3  strokes  are  here  ? 

Head,  supplying  the  missing  numbers  : 

4+-=6            G-4  =  .  6-.   =4 

2+.=6            6-2  =  .  6-.   =2 

3+.=6            6-3  =  .  r,  -.=3 

How  many  squares  are  here  ?   [__]  | j  LJ  Lj  L_ J  LJ  LJ 

How  many  more  than  5  squares  are  7  s(]uares  ? 
How  many  squares  must  you  put  with  two  squares  to 
make  7  squares  ? 


ADDITION  AND  SUBTRACTION.  1 1 

If  you  take  away  4  squares,  how  many  will  be  left  ? 
How  many  more  than  3  squares  are  7  squares  ? 
Read,  supplying  the  missing  numbers  : 

5+-=7  7-5=.  7-.  =5 

2  +  .  =7  7-2  =  .  7_  .  =2 

4+  .  =7  7-4=  .  7-  .  =4 

3+.  =7  7-3=.  7-.   =3 

How  many  rings  are  here  ?       OOOOOOOO 

How  many  more  than  6  rings  are  8  rings  ? 

How  many  rings  must  you  put  with  2  rings  to  make  8 

rings  ? 
If  you  take  3  rings  from  8  rings,  how  many  will  be  left  ? 
How  many  more  than  3  rings  are  8  rings  ? 
How  many  rings  must  you  put  with  4  rings  to  make  8 

rings  ? 

Read,  supplying  the  missing  numbers  : 

6+.=8  4+.=8  8-3  =  . 

2+.=S  8-6=.  8-4=. 

5+.=8  8-2  =  .  8-.  =5 

3+.=8  8-5=.  8  —  .=3 

How  many  crosses  are  here  ?         XXXXXXXXX 

How  many  more  than  7  crosses  are  here  ? 

How  many  crosses  must  you  put  with  2  crosses  to  make  9 

crosses  ? 
If  you  take  6  crosses  from  9  crosses,  how  many  will  be 

left? 
How  many  more  than  3  crosses  are  9  crosses  ? 


12   WHOLE  NUMBERS  AND  FRACTIONAL  PARTS 

How  many  more  than  5  crosses  are  9  crosses  ? 

If  5  crosses  are  taken  from  9  crosses,  how  many  are  left  ? 

Read,  supplying  the  missing  numbers  : 


7  +  .  =9 

9-7  = 

9-  . 

.  =7 

2+  .   =9 

9-2  = 

9  - 

.  =2 

6  +  .  =9 

9-6  = 

9- 

.  =  6 

3  4-  •   =  9 

9-3  = 

9- 

.  =3 

5  +  .   =  9 

9-5  = 

9- 

.  =  5 

4+  .  =9 

9-4=  . 

9  -  . 

=  4 

Forming,  Writing,  and  Reading  Numbers  from  10 
to  20;  Addition  and  Subtraction. 

18.  Oral  Exercise. 

How  many  are  9  and  1  ? 
Ten  ones  make  a  ten. 
Ten  is  written  10. 


How  many  are  10  and  1  ? 
Eleven  is  written  11. 


How  many  are  10  and  2  ? 
Twelve  is  written  12. 


How  many  are  10  and  3  ? 
Thirteen  is  written  13. 


How  111  any  are  10  and  4  ? 
Fourteen  is  written  14. 


NUMBERS  FROM  10  TO  20. 


13 


How  many  are  10  and  5  ? 
Fifteen  is  written  15. 

How  many  are  10  and  6  ? 
Sixteen  is  written  16. 


How  many  are  10  and  7  ? 
Seventeen  is  written  17. 

How  many  are  10  and  8  ? 
Eighteen  is  written  18. 

How  many  are  10  and  9  ? 
Nineteen  is  written  19. 

19.  Oral  Exercise. 

Read,  supplying  the  missing  numbers  : 

2  beans  and  8  beans  are  —  beans. 

4  pears  and  6  pears  are  —  pears. 

3  cherries  and  8  cherries  are  —  cherries. 

5  berries  and  5  berries  are  —  berries. 

4  lemons  and  7  lemons  are  —  lemons. 

5  figs  and  6  figs  are  —  figs. 

3  pies  and  7  pies  are  —  pies. 

2  apples  and  9  apples  are  —  apples. 
5  grapes  and  7  grapes  are  —  grapes. 

4  plums  and  8  plums  are  —  plums. 

5  rings  and  8  rings  are  —  rings. 

6  pens  and  6  pens  are  —  pens. 


14  WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 

3  cakes  and  9  cakes  are  —  cakes. 

6  rolls  and  8  rolls  are  —  rolls. 

5  hawks  and  9  hawks  are  —  hawks. 

7  crabs  and  8  crabs  are  —  crabs. 

4  rails  and  9  rails  are  —  rails. 

6  posts  and  7  posts  are  —  posts. 

8  boards  and  9  boards  are  —  boards. 

6  screws  and  9  screws  are  —  screws. 

7  saws  and  7  saws  are  —  saws. 

8  houses  and  8  houses  are  —  houses. 
7  cans  and  9  cans  are  —  cans. 

9  strings  and  9  strings  are  —  strings. 
9  caps  and  10  caps  are  —  caps. 


20.  Exercise. 

Copy,  writing  the  proper  figures  in  place  of  the  dots 


9  +  1  = 

6  +  4  = 

8  +  3  = 

9  +  3  = 

6  +  6  = 

7  +  0  = 
7  +  7  = 
9  +  7  = 
9  +  9  = 


8  +  2  = 
5  +  5  = 

7  +  4  = 

8  +  4  = 

9  +  4  = 
9  +  5  = 
9  +  6  = 
8  +  8  = 

10  +  8  = 


7  +  3  = 
9  +  2  = 

6  +  5  = 

7  +  5  = 

8  +  5  = 
8  +  6  = 

8  +  7  = 

9  +  8  = 
10  +  9  = 


21.  Oral  Exercise. 

Read  Exercise  20,  supplying  the  missing  numbers. 


ADDITION  AND  SUBTRACTION. 


15 


22.  Oral  Exercise. 

1  star  and  how  many  stars  are  10  stars  ? 

5  buttons  and   how  many  buttons  are   10 
buttons  ? 

2  mugs  and  how  many  mugs  are  10  mugs  ? 
4  boxes  and  how  many  boxes  are  10  boxes  ? 

3  dimes  and  how  many  dimes  are  10  dimes  ? 


10 


2  bricks  and   how  many  bricks   are   11 
bricks  ?  n 

5  ribbons  and  how  many  ribbons  are  11  ribbons  ? 

3  tacks  and  how  many  tacks  are  11  tacks  ? 

4  tubs  and  how  many  tubs  are  11  tubs  ? 


3  ropes  and   how  many  ropes   are   12 
ropes  ? 

6  caps  and  how  many  caps  are  12  caps  ? 

4  rugs  and  how  many  rugs  are  12  rugs  ? 

5  lamps  and  how  many  lamps  are  12  lamps  ? 


•  •••• 


•  • 


12 


4  globes  and  how  many  globes  are  13  ,  •©••• 

globes  ?  ' — 


•  •« 


6  maps  and  how  many  maps  are  13  maps  ? 
5  slates  and  how  many  slates  are  13  slates  ? 


13 


•  •••• 

•  •••• 


14 


5  pencils  and  how  many  pencils  are 
14  pencils  ? 

7  crayons  and  how  many  crayons  are  14  crayons  ? 

6  chains  and  how  many  chains  are  14  chains  ? 


•  ••• 


16    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 


6  blocks  and  how  many  blocks 
are  15  blocks  ? 

7  buttons  and  how  many  buttons 
are  15  buttons  ? 


15 


•  •••• 

•  •••• 


16 


7  hooks  and  how  many  hooks  are 

16  hooks  ? 

8  papers   and   liow  many  papers  are  16  papers  ? 

8  plants  and  how  many  plants  are     •••••     •• 

17  plants  ? 


••••• 


17 


9  acorns  and  how  many  acorns  are 
18  acorns  ? 

10  chestnuts  and  how  many  chest- 
nuts are  18  chestnuts  ? 

10  walnuts  and  how  many  walnuts 
are  19  walnuts  ? 


•  •••• 

•  •••• 


•  •• 

•  •••• 


18 


•  •••• 

•  •••• 


•  ••• 


19 


23.  Exercise. 

Copy,  writing  the  proper  figures  in  place  of  dots 


9  4- 

•  =  10 

9  +  • 

=  12 

8  +  .  =  14 

5  +  . 

►  =  10 

G  +  . 

=  12 

9  +  •  =  15 

8+  . 

.  =  10 

8  +  . 

=  12 

8  +  •  =  15 

6  +  « 

.  =10 

7  -f  • 

=  12 

9  +  .  =  16 

7  + 

.  =  10 

0  +  . 

=  13 

8  +  •  =  16 

9  + 

.  =  11 

7  +  . 

=  13 

9  +  .  =  17 

6  + 

.  =  11 

8  +  . 

=  13 

9  +  .  =  18 

8  + 

►  =  11 

9  +  • 

=  14 

8  +  •  =  18 

7  +  • 

►  =  11 

7+  . 

=  14 

9  +  .  =  19 

ADDITION.  17 


24.  Oral  Exercise. 

Read  Exercise  23,  supplying  the  missing  numbers. 

25.  Oral  Exercise. 

flow  many  melons  and  5  melons  are    , 

10  melons  ?  ^0 

How  many  lemons  and  2  lemons  are  10  lemons  ? 
How  many  shrubs  and  4  shrubs  are  10  shrubs  ? 
How  many  baskets  and  3  baskets  are  10  baskets  ? 


How  many  dots  and  2  dots  are  11  dots  ?[»#»»»»#♦»» 
How   many  trees   and    5  trees  are  11  11 

trees  ? 
How  many  ducks  and  4  ducks  are  11  ducks? 
How  many  geese  and  3  geese  are  11  geese  ? 

How  many  bears  and  3  bears  are  12      #• 

bears  ?  • 


12 

How  many  camels  and  G  camels  are  12  camels  ? 

How  many  monkeys  and  4  monkeys  are  12  monkeys  ? 
How  many  lions  and  5  lions  are  12  lions  ? 


•  •0 


How  many  deer  and  4  deer  are  13 

deer?  |  ••••••••*>1 

How  many  goats  and  6  goats  are  13  goats  ? 
How  many  rabbits  and  5  rabbits  are  13  rabbits  ? 

How  many  owls  and  7  owls  are  14     i  ##^#####«0  i 


owls  ?  14 

llow  many  bats  and  5  bats  are  14  bats  ? 
How  many  minks  and  G  minks  are  14  minks  ? 


18    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 


How  many  flags  and  6  flags  are  15 
flags  ? 


15 


IIow  many  hoops  and  8  hoops  are  15  hoops  ? 


How  many  wheels  and  8  wheels  arc 
10  wheels  ? 


16 


How  many  spokes  and  7  spokes  are  IG  spokes  ? 


How  many  trout  and  8  trout  are  17     | — 

trout  ? 
How  many  frogs  and  7  frogs  are  17  frogs 


17 


How  many  shingles  and  8  shingles     r— - 

are  18  shingles  ? 
How  many  nails  and  9  nails  are  18  nails  ? 

How  many  wheels  and  9  wheels  are         i 


18 


19  wheels  ? 


19 


26.  Exercise. 

Copy,  writing  the  proper  figures  in  place  of  dots  : 


.  +  9  =  10 

►  +  9  =  12 

.  +  8  =  14 

.  +  5  =  10 

.  +  6  =  12 

.  +  9  =  15 

.  +  8  =  10 

+  8=12 

.  +  8  =  15 

.  +  6  =  10 

+  7  =  12 

+  9  =  16 

.  +  7  =  10 

+  9  =  13 

+  8=  16 

.  +  9  =  11 

+  7  =  13 

•  +  9  =  17 

.  +  6  =  11 

+  8  =  13 

+  9  =  18 

.  +8  =  11 

.  +  9  =  14 

+  10  =  18 

.  +  7  =  11 

+  7  =  14 

+  9  =  19 

SUBTRACTION. 


19 


27.  Oral  Exercise. 

Bead  Exercise  26,  supplying  tlie  missing  numbers. 

28.  Oral  Exercise. 

Read,  supplying  the  missing  numbers  : 
10  horses  less  1  horse  are  —  horses. 
10  cows  less  6  cows  are  —  cows. 
10  calves  less  8  calves  are  —  calves. 
10  mules  less  2  mules  are  —  mules. 
10  sheep  less  7  sheep  are  —  sheep. 
10  goats  less  4  goats  are  —  goats. 
10  pigs  less  9  pigs  are  —  pigs. 
10  birds  less  5  birds  are  —  birds. 

10  geese  less  3  geese  are  —  geese. 

11  ducks  less  2  ducks  are  —  ducks. 
11  carts  less  5  carts  are  —  carts. 
11  coats  less  8  coats  are  —  coats. 
11  vests  less  3  vests  are  —  vests. 
11  hats  less  6  hats  are  —  hats. 
11  caps  less  4  caps  are  —  caps. 
11  collars  less  7  collars  are  —  collars. 

11  buttons  less  9  buttons  are  —  buttons, 

12  pins  less  3  pins  are  —  pins. 
12  needles  less  6  needles  are  —  needles. 
12  skates  less  8  skates  are  —  skates. 
12  plows  less  5  plows  are  —  plows. 
12  wheels  less  4  wheels  are  —  wheels. 
12  rings  less  7  rings  are  —  rings. 
12  cuffs  less  9  cuffs  are  —  cuffs. 


•  •••• 

•  •••• 


•  • 


20    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 


13  matches  less  4  matches  are  —  matches. 
13  lamps  less  6  lamps  are  —  lamps. 
13  cars  less  8  cars  are  —  cars. 
13  boys  less  5  boys  are  —  boys. 
13  girls  less  7  girls  are  —  girls. 

13  women  less  9  women  are  —  women 

14  dogs  less  5  dogs  are  —  dogs. 
14  cats  less  8  cats  are  —  cats. 
14  mice  less  6  mice  are  —  mice. 
14  birds  less  9  birds  are  —  birds. 

14  flowers  less  7  flowers  are  —  flowers, 

15  houses  less  6  houses  are  —  houses. 

15  boards  less  8  boards  are  —  boards.    

15  shingles  less   9  shingles  are  —  shingles. 

15  bricks  less  7  bricks  are  —  bricks. 

16  slates  less  7  slates  are  —  slates. 
16  pencils  less  9  pencils  are  —  pencils. 

16  buds  less  8  buds  are  —  buds. 

17  twigs  less  8  twigs  are  —  twigs. 

17  trees  less  9  trees  are  —  trees. 

18  shrubs  less  9  shrubs  are  —  shrubs. 


•  •••• 

•  •••• 


•  •• 


•  •••• 

•  •••• 


•  •••• 

•  •••• 


•  •••• 

•  •••• 


•  •• 


••• 


•  •••• 

•  •••• 


•  •• 

•  ••• 


29.  Exercise. 

Copy,  writing  the  proper  figures  in  place  of  dots  : 

10  -  1  =   .  10  -  7  =  •  11  -  9  = 

10-9=-  10-4=.  11-3  = 

10-2=.  10-6=.  11-8  = 

10-8=.  10-5=.  11-4  = 

10-3=.  11-2=.  11-7  = 


ADDITION 

21 

11-5  = 

13  -  9  = 

15  -  6  =  • 

11  -  6  =  < 

13  -  5  = 

15  -  9  =  « 

12-3=  ^ 

13  -  8  =  < 

15  -  7  =  . 

12  -  9  =  . 

13  -  6  =  . 

15  -  8  =  ' 

12-4=  . 

13  -  7  = 

16  —  7=  . 

12  -  8  =  . 

14  -  5  =  . 

16  -  9  =  « 

12  -  5  =  . 

14  -  9  = 

16  -  8  =  . 

12-7=  . 

14  -  6  =  . 

17-8=  . 

12-6  = 

14  -  8  =  « 

17-9=  . 

13  -  4  =  . 

14  -  7  =  • 

18  -  9  =  . 

30.  Oral  Exercise. 

Eead  Exercise  29,  supplying  the  missing  numbers. 

31.  If  we  count  3  to  4,  we  have  7. 
Counting  3  to  4  is  called  adding  3  to  4. 


32.  Oral  Exercise. 

2        4        13 


9 


To  each  of  the  numbers  above  the  line: 

/.   Add  1.  Thus,  2  and  1  are  3,  4  and  1  are  5,  and  so  on. 

2.  Add  2.         4.  Add  4.  6.  Add  6.  8.  Add  8. 

3.  Add  3.         5.  Add  5.  7.  Add  7.  9.  Add  9. 

33.  Oral  Exercise. 

To  the  teacher. — Teach  Roman  notation  to  XIX.     Teach  United 
States  coin  to  one  dollar. 

/.  Charles   had   2  goldfish  and  his  uncle  gave  him  3 
more  ;  how  many  had  he  then  ? 


22   WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 

2.  Sara  found  3  eggs  in  one  nest  and  4  in  another  ;  how 
many  did  she  find  in  all  ? 

3.  Mr.  Rice  has  2  sons  and  4  daughters ;  how  many 
children  has  he  ? 

4.  A  farmer  had  4  sheep  and  bought  5  more ;  how 
many  had  he  then  ? 

5.  James  is  2  years  older  than  Mary ;  if  Mary  is  6 
years  old,  how  old  is  James  ? 

6.  How  many  lambs  had  a  farmer  at  first,  if  he  had  5 
left  after  selling  6  ? 

7.  Jane  spelled  6  words  correctly  and  missed  4 ;  how 
many  was  she  given  to  spell  ? 

8.  How  many  horses  has  a  man,  if  he  keeps  3  to  drive 
and  6  to  work  ? 

9.  Harry  has  walked  4  miles  toward  home  and  has  7 
miles  yet  to  go  ;  how  far  from  home  was  he  when  he 
started  ? 

10.  How  many  animals  did  a  blacksmith  shoe  to-day,  if 
he  shod  7  horses  and  5  mules  ? 

//.  How  many  boys  were  there  at  first  in  a  class,  if  7  re- 
mained after  3  were  promoted  ? 

12.  How  many  policemen  must  be  on  duty  so  that  when 
4  go  home  8  remain  ? 

10  cents  make  1  dime. 

13.  1  cent  and  how  many  cents  make  a  dime  ? 

14.  A  dime  is  how  much  more  than  a  5-cent  piece  ? 

15.  A  dime  and  a  cent  make  how  much  ? 

16.  A  dime  and  a  5-cent  piece  make  how  much  ? 


ADDITION  AND  SUBTRACTION.  23 

17.  A  dime,  a  cent,  and  a  5-cent  piece  make  how 
much  ? 

18.  A  dime  is  worth  how  many  5-cent  pieces  ? 

19.  A  cent  and  a  5-cent  piece  are  together  worth  how 
much  less  than  a  dime  ? 

34.  Exercise. 

To  the  teacher.  — Let  the  pupil  solve  problems  in  Exercises  orally 
when  possible. 

/.  There  are  3  persons  in  a  boat  that  will  carry  8  per- 
sons ;  how  many  more  may  get  in  ? 

2.  I  found  6  eggs  ;  how  many  more  must  I  get  to  have 

12  ? 

7  days  mahe  1  week. 

3.  Name  the  days  of  the  week. 

4.  Name  the  first  day  of  the  week  ;  the  third ;  the 
second  ;  the  fourth  ;  the  sixth  ;  the  fifth ;  the  seventh. 

5.  Last  week  father  worked  4  days  ;  how  many  days 
was  he  idle  ? 

6.  A  man  owns  6  horses  ;  how  many  more  must  he  buy 
to  have  14  ? 

7.  Samuel  has  spent  2  days  with  his  cousin  ;  how  long 
has  he  yet  to  stay,  if  his  visit  is  to  last  a  week  ? 

8.  Mother  put  up  17  cans  of  fruit  to-day ;  8  of  them 
were  pears  and  the  others  peaches ;  how  many  were 
peaches  ? 

9.  I  had  to  write  16  words  ;  I  have  written  7  of  them ; 
how  many  have  I  yet  to  write  ? 


24   WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 

10.  My  uncle  wishes  to  have  12  cows ;  he  now  has  5  ; 
how  many  more  must  he  get  ? 

11.  K  boy  paid  10  cents  for  a  loaf  of  bread  and  a  pound 
of  sugar.  If  the  sugar  cost  6  cents,  what  did  the  loaf  of 
bread  cost  ? 

12.  How  many  days  of  the  week  are  left  after  Wednes- 
day has  passed  ? 

35.  If  we  take  3  from  7,  we  have  4. 

Taking  3  from  7  is  called  subtracting  3  from  7. 

36.  Oral  Exercise. 

Subtract : 
/.  2  from 

2  4         3         O         9         5         8       10         7       11 

Thus,  2  from  2  leaves  0,  2  from  4  leaves  2,  and  so  on, 

2.  3  from 

3  6    4    9    5    8   12   lO    7   11 

3.  4  from 

7  5    6    4   12    8   11    9   13   10 

4.  5  from 

8  5    O   12   14   11    9   13   lO    7 

6.   6  from 

8   10    6    9   12    7   14   11   13   15 

6.  7  from 

8   lO    9    7   12   14   11   16   13   15 

7.  8  from 

10  12   14   16    8    9   11   13   16   17 

8.  9  from 

11  13    9   12   16   10   14   17   15   18 


ADDITION  AND  SUBTRACTION.  25 

37.  Oral  Exercise. 

/.  Mary  saw  8  stars,  but  a  cloud  now  covers  3  of  them  ; 
how  many  can  she  still  see  ? 

2.  There  were  7  houses  in  a  row,  but  a  fire  burned  3  of 
them  ;  how  many  were  then  left  ? 

3.  A  gardener  planted  9  trees,  but  4  of  them  died  ;  how 
many  lived  ? 

4.  A  farmer  had  11  colts,  but  sold  5  of  them  ;  how 
many  had  he  left  ? 

5.  9  boys  were  examined ;  if  5  of  them  failed,  how 
many  passed  ? 

6.  Amos  had  10  problems  to  solve  ;  if  he  has  solved  3, 
how  many  has  he  yet  to  solve  ? 

7.  There  were  12  women  in  a  parlor,  but  4  of  them 
went  away  ;  how  many  remained  ? 

8.  A  woman  bought  12  eggs,  and  cooked  6  of  them  for 
supper  ;  how  many  had  she  left  ? 

12  is  called  a  dozen  (doz.). 

9.  A  man  bought  a  dozen  oranges  and  ate  3  of  them  ; 
how  many  had  he  left  ? 

10.  A  lady  bought  a  dozen  ears  of  corn  and  cooked  8  of 
them  for  dinner  ;  how  many  had  she  left  ? 

//.  James  planted  15  ferns,  but  only  9  of  them  lived  ; 
how  many  died  ? 

12.  A  boy  had  10  firecrackers  and  shot  off  6  of  them  ; 
how  many  had  he  then  left  ? 


26    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 

33.  Oral  Exercise. 

/.  A  farmer  had  9  lambs ;  if  he  sold  all  but  2,  how 
many  did  he  sell  ? 

2.  A  newsboy  bought  15  morning  papers  and  sold  all 
but  6  ;  how  many  did  he  sell  ? 

3.  Jessie  bought  a  pound  of  sugar  and  gave  the  store- 
keeper 10  cents.  He  gave  her  4  cents  change  ;  how  much 
did  he  charge  for  the  sugar  ? 

4.  There  are  4  collars  left  in  a  box  that  had  contained 
a  dozen  ;  how  many  were  taken  out  ? 

5.  14  persons  came  into  the  dining  room  for  dinner ; 
there  is  room  for  only  9  ;  how  many  must  wait  ? 

6.  Frank  has  6  cents  left  out  of  14  cents  that  his  father 
gave  him  ;  he  spent  the  rest  for  a  tablet ;  how  much  did 
it  cost  ? 

7.  Chester  had  13  marbles,  but  he  lost  all  but  4 ;  how 
many  did  he  lose  ^ 

8.  How  many  shad  did  a  man  sell,  if  he  had  a  dozen 
and  sold  all  but  3  ? 

9.  Last  spring  I  planted  12  shade  trees  ;  they  all  died 
but  4  ;  how  many  died  ? 

10.  K  boy  had  11  pigeons  and  sold  all  but  4  ;  how  many 
did  he  sell  ? 

//.  Frank  had  a  dozen  plums  ;  he  kept  4  for  his  sister 
and  ate  the  others  ;  how  many  did  lie  eat  ? 

12.  I  have  5  cents  yet  to  get  in  order  to  have  14  cents ; 
how  much  money  have  I  ? 


ADDITION  AND  SUBTRACTION.  27 

39.  Oral  Exercise. 

/.  6  pupils  are  absent  from  a  class  and  9  are  present ; 
how  many  belong  to  this  class  ? 

2.  A  man  was  sent  to  prison  5  years  ago  and  he  has  yet 
7,  years  to  serve  ;  how  many  years  must  he  serve  in  all  ? 

3.  There  were  12  knives  on  a  table,  but  only  9  forks  ; 
how  many  more  knives  than  forks  were  there  ? 

4.  A  farmer  had  8  sheep  left  after  selling  7  ;  how  many 
had  he  at  first  ? 

5.  There  were  16  tramps  in  a  workhouse,  but  7  of  them 
were  set  free  ;  how  many  were  held  ? 

6.  How  many  girls  were  in  a  class  at  first,  if  7  remained 
after  6  were  promoted  ? 

7.  How  many  cents  must  I  put  in  my  money  bank  to 
have  14  cents  in  it,  if  it  already  contains  9  cents  ? 

8.  I  have  put  4  cans  of  corn  into  a  box  that  will  hold 
a  dozen  cans  ;  how  many  more  will  it  hold  ? 

9.  There  are  14  sheep  and  lambs  in  a  field ;  how  many 
of  them  are  lambs,  if  there  are  6  sheep  ? 

10.  How  many  men  must  my  father  hire  to  have  14,  if 
he  already  has  7  ? 

//.  After  a  fire  had  burned  8  stores  in  a  block,  there 
were  9  left ;  how  many  were  there  before  the  fire  ? 

12.  A  girl  bought  a  pound  of  sugar  for  6  cents  and  re- 
ceived 4  cents  change  ;  how  much  money  did  she  give  the 
grocer  ? 


28    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 

Forming,  Writing,  and  Reading  Numbers  from  20 
to  lOO;    Addition  and  Subtraction. 

40.  Two  tens  make  twenty ,  written  20. 

Twenty 


Two  tens  and  one  make  twenty-one,  written  21. 

Twenty-one 

Two  tens  and  two  make  tiventy-two,  written  22. 

Twenty-two 


Two  tens  and  three  make  twenty-three,  writ- 
ten 23. 


ij 

Twenty-three 

1!" 


Twenty  and  four  make  twenty-four,  written  24. 

Twenty-four 

Twenty  and  five  make  twenty -five,  written  25. 

Twenty-five 

Twenty  and  six  make  twenty-six,  written  26/ 


Twenty   and  seven   make   tiventy-seven, 
written  27. 


Twenty-six 


TTll 


Twenty-seven 


NUMBERS  FROM  20  TO   100. 


29 


Twenty  and   eight   make   twenty -eight, 

written  28.  ^  ^ 

Twenty-eight 

Twenty   and   nine    make    twenty -nine,     ffll  ffl  1 1 1 1  j  1 1 

written  29.  li  11 1  I  I  1 1 1  J 

Twenty-nine 

41.  Oral  Exercise. 

Beginning  at  1,  count  to  29. 

Read  these  numbers  :  20,  21,  22,  23,  24,  25,  26,  27,  28,  29. 

Tell  the  number  of  tens  and  ones  in  each  of  the  above 

numbers. 
How  many  ones  are  there  in  2  tens  and  4  ?    2  tens  and  6  ? 

Three  Tens 


42.  Three  tens  make  thirty,  written  30. 


Three   tens  and   one  make   thirty-one,   writ- 
ten 31. 

43.  Oral  Exercise. 

Three  tens  and  two  make  what  number  ? 
Three  tens  and  three  make  what  number  ? 
Three  tens  and  four  make  what  number  ? 
Three  tens  and  five  make  what  number  ? 
Three  tens  and  six  make  what  number  ? 
Three  tens  and  seven  make  what  number  ? 
Three  tens  and  eight  make  what  number  ? 
Three  tens  and  nine  make  what  number  ? 


Thirty-one 


30    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 


Beginning  at  30,  count  to  30. 

Read  these  numbers  :  30,  31,  32,  33,  34,  35,  36,  37,  38,  39. 

Tell  the  number  of  tens  and  ones  in   each   of  the  above 

numbers. 
How  many  ones  are  there  in  3  tens  and  1  ?    3  tens  and  6  ? 

Four  Teng 
1?^ 


44.  Four  tens  imxkQ  forty,  written  40. 


-1 
i 


Forty 


45.  Oral  Exercise. 

Four  tens  and  one  make  what  number  ? 

Beginning  at  forty,  name  the  numbers  that  are  formed 
by  forty  and  one  ;  forty  and  two  ;  forty  and  three  ;  forty 
and  four  ;  forty  and  five ;  forty  and  six  ;  forty  and  seven; 
forty  and  eight ;  forty  and  nine. 

Read  :  40,  41,  42,  43,  44,  45,  46,  47,  48,  49. 


Beginning  at  forty,  count  to  forty-nine. 
46.  Five  tens  make  Jifii/,  written  50. 

Six  tens  make  sixty,  written  60. 


Seven  tens  make  seventy,  written  70. 


Eight  tens  make  eighty,   writ- 
ten 80. 


Five  Tens 


itj 


ui 


Fifty 
Six  Tena 


Eighty 


NUMBERS  FROM  20  TO  100. 


31 


Nine  Tens 


Nine  tens  make  ninety,  written 
90. 


Ninety 

Ten  Tens 


Ten  tens  combined  make  one  hundred, 
written  100. 

One  Hundred 

47.  Oral  Exercise. 

/.  Name  the  numbers  from  fifty  to  sixty. 

2.  Name  the  numbers  from  sixty  to  seventy. 

3.  Name  the  numbers  from  seventy  to  eighty. 

4.  Name  the  numbers  from  eighty  to  ninety. 

5.  Name  the  numbers  from  ninety  to  one  hundred. 

6.  Pvead  :  50,  51,  52,  53,  54,  55,  56,  57,  58,  59. 

7.  Read  :  60,  61,  62,  63,  64,  65,  66,  67,  68,  69. 

8.  Read  :  70,  71,  72,  73,  74,  75,  76,  77,  78,  79. 

9.  Real!  :  80,  81,  82,  83,  84,  85,  86,  87,  88,  89. 
10.  Read  :  90,  91,  92,  93,  94,  95,  96,  97,  98,  99. 


48.  Exercise. 

Write  in  fiofures  the  numbers  : 


/.  From  one  to  ten. 

2.  From  ten  to  twenty. 

3.  From  twenty  to  thirty. 

4.  From  thirty  to  forty. 

5.  From  forty  to  fifty. 


6.  From  fifty  to  sixty. 

7.  From  sixty  to  seventy. 

8.  From  seventy  to  eighty. 

9.  From  eighty  to  ninety. 
10.  From  ninety  to  one  hun- 
dred. 


5. 

32 

9. 

51 

13. 

44 

17. 

72 

21.   91 

6. 

34 

10. 

57 

14. 

46 

18. 

70 

22.   19 

7. 

36 

11. 

20 

15. 

64 

19. 

15 

23.   92 

8. 

39 

12. 

33 

16. 

69 

20. 

88 

24.  87 

32    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 

49.  Oral  Exercise. 

Read  : 
/.  22 
2.  25 
5.  28 
4.  30 

50.  Oral  Exercise. 

/.  Count  from  10  to  20. 

2.  Count  backwards  from  20  to  10. 

3.  Count  from  20  to  30. 

4.  Count  backwards  from  30  to  20. 

5.  Count  from  30  to. 40. 

6.  Count  backwards  from  50  to  40. 

7.  Count  from  50  to  60. 

8.  Count  backwards  from  70  to  60. 

9.  Count  from  70  to  80. 

10.  Count  backwards  from  90  to  80. 

51.  In  numbers  of  two  figures,  the  right-hand 
figure  is  called  the  Units'  figure,  and  the  left-hand 
figure  the  Tens'  figure. 

52.  Oral  Exercise. 

/.  In  87  name  the  tens'  figure  ;  the  units'  figure. 

2.  In  63  how  many  tens  are  there,  and  how  many 
units  ? 

3.  In  90  how  many  tens  are  there,  and  how  many  units  ? 

4.  How  many  figures  are  needed  to  write  a  number 
mailc  uj)  of  3  tens  and  2  units  ? 


ADDITION  AND  SUBTRACTION.  33 

5.  Name  the  number  made  up  of  6  tens  and  2  units. 

6.  What  is  the  largest  number  that  can  be  written  with 
one  figure  ?     With  two  figures  ? 

7.  What  two  numbers  can  be  written  with  the  figures 
8  and  9  ? 

To  the  teacher.     Teach  Roman  notation  to  C.     See  p.  79. 

53.  Oral  Exercise. 

/.  Add  each  of  the  numbers  2,  5,  3,  6,  4,  7,  9,  8,  10  to 

10  20       30        40       50       60        70        80       90 

Name  results  only.      Thus,   12,  15,  13,  and  so  on  ;  then  22,  25, 
23,  and  so  on  ;  etc. 

2.  Add  each  of  the  numbers  2,  5,  3,  6,  4,  8,  7  to 

11  21        31        41        51        61        71        81        91 

3.  Add  each  of  the  numbers  2,  5,  3,  6,  4,  7  to 

12  22        32        42       52       62        72        82        92 

4.  Add  each  of  the  numbers  2,  5,  3,  G,  4  to 

13  23       33       43       53       63        73        83       93 

5.  Add  each  of  the  numbers  2,  5,  4,  3  to 

14  24        34        44        54        64        74        84        V\ 

6.  Add  each  of  the  numbers  2,  4,  3  to 

15  25       35        45        5^        65        75        85        95 

7.  Add  each  of  the  numbers  3  and  2  to 

16  26   36   46   56   66   76   86   96 

8.  Add  2  to 

17  27   37   47   57   67   77   87   97 

8 


34   WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 


64.  Oral  Exercise. 

Note.     The  sign  ?  denotes  how  many? 

/.    74.3_|_4_5=?  Answer  thus  :  7,  10,  14,  0. 

2.  3+4  +  2=?  7.     5  +  4  +  G  +  3=? 

3.  7+5  +  6=?  8.     7  +  4  +  3-5=? 

4.  2  +  5-1  =  ?  9.     9-4  +  6-7=? 

5.  14  -  9  +  7  =  ?  /(?.  14  -  9  +  8  -  6  =  ? 
ff.   18  -  9  -  4  =  ?  //.  18  -  9  -  3  -  4  =  ? 

55.  Exercise. 

/.  I  bought  a  tablet  for  4  cents,  a  pencil  for  5  cents, 
and  a  sponge  for  6  cents  ;  how  much  did  I  pay  for  all  ? 

2.  There  are  4  chairs  at  each  side  of  a  table  and  1  at 
each  end  ;  how  many  are  there  in  all  ? 

3.  A  bicycle,  a  cart,  and  a  wagon  have,  in  all,  how 
many  wheels  ? 

4.  From  a  box  of  1  dozen  collars  I  sold  6  to  one  man 
and  3  to  another  ;  how  many  remain  ? 

5.  A  man  owed  12  dollars.  lie  paid  4  dollars  at  one 
time  and  3  dollars  at  another  ;  how  much  does  he  still  owe  ? 

6.  A  boy  added  three  numbers,  and  the  answer  was  10. 
If  two  of  the  numbers  were  3  and  5,  what  was  the  third  ? 

7.  2  sheep,  4  cows,  2  horses,  and  2  mules  make  how 
many  head  of  stock  ? 

8.  A  laborer  had  5  dollars  left  of  his  month's  wages 
after  paying  4  dollars  house  rent  and  9  dollars  store  bill ; 
what  were  his  wages  ? 

9.  Out  of  a  dozen  eggs  mother  used  2  for  breakfast,  4 
for  dinner,  and  4  for  supper  ;  liow  many  were  left  ? 


ADDITION  AND  SUBTRACTION.  35 

10.  6  women,  5  girls,  3  men,  and  2  boys  are  how  many 
persons  ? 

//.I  have  4  brothers  and  3  sisters  ;  how  many  children 
are  there  in  our  family  ? 

12.  Mother  sent  me  to  the  store  for  a  5-cent  spool  of 
thread,  a  4-cent  loaf  of  bread,  and  a  2-cent  box  of  matches. 
How  much  change  should  I  bring  back  to  her  out  of  a 
dime  and  a  5-cent  piece  ? 

66.  Oral  Exercise. 

/.  Add  each  of  the  numbers  2,  5,  3,  6,  4,  8,  7,  9  to 
19        29       39       49       59        69        79        89 

Name  results  only.  Thus,  31,  24,  23,  and  so  on  ;  then  31,  34, 
33,  and  so  on  ;  etc. 

2.  Add  each  of  the  numbers  2,  5,  3,  0,  4,  8,  7,  9  to 
18       38        58        28        48        68        88        78 

3.  Add  each  of  the  numbers  5,  3,  6,  4,  8,  7,  9  to 
17        37        57        27        47        67        87        77 

4.  Add  each  of  the  numbers  5,  8,  6,  9,  7,  4  to 

16       36       56        26       46    .  66        86        76 

5.  Add  each  of  the  numbers  5,  8,  G,  7,  9  to 

15        35        55        25        45        65        85        75 

6.  Add  each  of  the  numbers  G,  8,  7,  9  to 

14        34        54        24        44        64        84        74 

7.  Add  each  of  the  numbers  8,  7,  9  to 

13       33       53        23       43        63        83        73 

8.  Add  each  of  the  numbers  8  and  9  to 

12   32   52   22   42   62   82   72 

9.  Add  9  to 

11   31   51   21   41   61   81   71 


36    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 

Forming,  Writing,  and  Reading  Numbers  from  1  GO 
to  lOOO;  Addition  and  Subtraction. 

57.  Two  hundred   is  written  200  ;  three   hundred, 
300  ;  nine  hundred,  900. 

58.  One  hundred  and  one  make  one  hundred  one, 
written  101. 

One  hundred  and  two   make  one   hundred  two, 

written  102. 
One   hundred   and  ten    make  one    hundred  ten, 

written  110. 
Two  hundred  and  forty  make  two  hundred  forty, 

written  240. 
Six  hundred  and  seventy-two  make  six  hundred 

seventy-two,  written  672. 
Nine  hundred  and  ninety-nine  make  nine  hundred 

ninety-nine,  written  999. 

59.  In  numbers  of  three  figures  the  first  from  the 
right  is  called  the  Units'  figure,  the  second  the  Tens* 
figure,  and  the  third  the  Hundreds*  figure. 

60.  Exercise. 
Write  in  figures  : 

/.  Four  hundred  ;  seven  hundred. 
2.   Five  liundred  ;  eight  hundred. 
5.  Six  hundred  ;  nine  hundred. 

4.  One  hundred  three  ;  one  liundred  eleven. 

5.  Two  liundred  twenty  ;  three  hundred  ninety-nine. 

6.  Four  hundred  four  ;  five  hundred  sixty-eight. 


NUMBERS  FROM    100   TO    1000. 


37 


7.  Six  hundred  sixteen  ;  seven  hundred  seventy-seven. 

8.  Nine  hundred  ninety-nine  ;  four  hundred  forty-four. 


61.  Oral  Exercise. 

Read   : 

/.  103. 

8.  213. 

15. 

900. 

22. 

527. 

29. 

149. 

2.  112. 

9.  400. 

16. 

901. 

23. 

752. 

30. 

222. 

3.  126. 

10.  407. 

17. 

910. 

24. 

257. 

31. 

806. 

4.  129. 

//.   510. 

18. 

888. 

25. 

600. 

32. 

419. 

5.  131. 

12.   111. 

19. 

808. 

26. 

999. 

33. 

344. 

6.  209. 

13.  222. 

20. 

919. 

27. 

909. 

34. 

707. 

7.  211. 

/4.   876. 

21. 

764. 

28. 

526. 

35. 

818. 

62.  Oral  Exercise. 

/.  Count  from  100  to  110. 

2.  Count  from  330  to  340. 

3.  Count  from  489  to  500. 

4.  Count  from  789  to  911. 

63.  Oral  Exercise. 

/.  In  765  how  many  units  ? 
hundreds  ? 

2.  In  860  how  many  units  ? 

hundreds  ? 

3.  In  409  how  many  units  ? 

hundreds  ? 


5.  Count  backwards  from 

210  to  200. 

6.  Count  backwards  from 

370  to  360. 

7.  Count  backwards  from 

910  to  890. 

8.  Count  backwards  from 

701  to  689. 

how  many  tens  ?  how  many 
how  many  tens  ?  how  many 
how  many  tens  ?   how  many 


38    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 

4.  llow  iiijuiy  figuroK  iiiu;  ii(3(3dod  to  write  u  number  made 

14)  of  3  hundreds,  2  tens,  and  5  units  ? 

5.  How  many  figures  are  needed  to  write  a  number  made 

up  of  4  hundreds  and  2  tens  ? 

6.  How  many  figures  are  needed  to  write  a  number  made 

up  of  2  hundreds  and  3  units  ? 

7.  Name  tlie  number  made  up  of  2  hundreds  and  2  tens. 

8.  What  is  the  largest  number  that  can  be  written  with 

three  figures  ?    What  is  the  smallest  ? 
To  the  teacher. — Teach  Roman  notation  to  M.     See  p.  79. 

64.  The  process  of  taking  two  or  more  numbers 
together  is  Addition. 

65.  The  answer  in  addition  is  called  the  Sum. 

66.  Add  427  and  532. 

2  units  and  7  units  are  9  units. 

427 

3  tens  and  2  tens  are  5  tens. 

- —         5  hundreds  and  4  hundreds  are  9  hundreds. 
Sum  959         „ 

The  sum  is  9o9. 

Tn  practice^  say  2  and  7  are  9  ;  S  and  2  are  5 ;  5  and  ^  are  9. 

67.  Exercise. 
Copy  and  add  : 

/.  28  3.  43  5.  50  7.  25 

JU  36^  32  43 

21 

2.  52  4.   05  6.  43  8.   10 

^  34  j^  52 

31 


ADDITION  AND  SUBTRACTION.  39 


9.   80 

13.   235 

17.   222 

21.   222 

11 

723 

333 

333 

7 

444 

10.   53 

14.   327 

18.   423 

22.  708 

23 

652 

322 

260 

13 

31 

//.  42 

15.   326 

/5.  321 

23.  421 

35 

362 

432 

243 

22 

146 

324 

12.   14 

/ff.  342 

20.   321 

24.   423 

32 

635 

372 

332 

23 

305 

143 

68.  The  process  of  taking  one  number  from  another 
is  Subtraction. 

69.  The  answer  in  subtraction  is  called  the  Differ- 
ence. 

70.  Subtract  252  from  594. 

2  units  from  4  units  leaves  2  units. 
Minuend       594         5  tens  from  9  tens  leaves  4  tens. 
Subtrahend  252         2  hundreds  from  5  hundreds  leaves  3  hun- 

Difference     342  ^reds. 

The  difference  is  342. 

In  practice,  say  2  from  4  haves  2 ;  5  from  9  leaves  J^  ;  2  from  5 
leaves  3. 


40    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 


71.  Exercise. 

Copy  and  subtract : 

/.  57 

7.  38 

13.   456 

19.   704 

23 

17 

234 

203 

2.   8G 

8.   26 

U.   397 

20.  420 

52 

4 

175 

310 

3.   65 

9.   48 

/5.  658 

21.   517 

32 

33 

423 

306 

4.   79 

10.   99 

16.   965 

22.   427 

_45 

66 

532 

120 

5.  59 

//.  87 

n.   878 

25.  575 

16 

75 

532 

475 

6.   97 

12.   74 

/S.  757 

24.   379 

6 

33 

243 

375 

72.  Add  76  and  38. 

8  units  and  6  units  are  14  units,  or  1  ten  and  4  units. 
76  Write  4  in  units*  place  and  add  1  to  the  tens.  1  ten, 
38       3  tens,   and  7  tens  are   11   tens,   or    1    hundred   and   1 


114       ten.     Write  1  in  tens'  place  and  1  in  hundreds'  place. 

The  sum  is  114. 
In  'practice,  say  8  and  6  are  14;  1,  3  and  7  are  11. 


73  Exercise. 

Copy  and  add  : 

/.  43 

2.   53 

3.   78 

4.   73 

48 

27 

51 

36 

ADDITION  AND  SUBTRACTION.  41 


5.   48 

/3.  89 

2/.  82 

29.     93 

85 

97 

20 

27 

19 

34 

55 

6.   79 

14.   98 

22.   58 

30.     75 

66 

43 

67 

89 

38 

32 

56 

7.   37 

/5.  39 

23.   27 

31.   185 

87 

93 

96 

237 

14 

8.   54 

16.   37 

24.   89 

52.  476 

98 

99 

9 

238 

17 

9.   76 

17.   54 

25.   37 

35.  567 

59 

77 

29 

196 

41 
23 

10.   27 

/5.  87 

26.   17 

34.   438 

98 

16 

29 

299 

51 
93 

//.  89 

19.   48 

27.   99 

35.   347 

7 

37 

9 

468 

55 

83 

20 

f2.   58 

20.   98 

28.   54 

36.   298 

^ 

32 

35 

386 

29 

27 

58 

42    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 


37.   275 

40.     74 

43.   127 

46.   175 

378 

186 

346 
428 

319 
416 

38.   G54 

41.   237 

44.   486 

47.   209 

196 

179 

201 

127 

302 

348 

39.   289 

42.     37 

45.   174 

45.  498 

69 

864 

297 

126 

488 

299 

74.  Subtract  37  from  65. 

7  units  cannot  be  taken  from  5  units.     Take  1  ten 

from  the  G  tens,  leaving  5  tens.     Add  this  1  ten, 
Mm.  65  .  ,  .       .  ^      .        ^       . 

o    1.    oi*y      or  10  units,  to  5  units,  making  15  units.     7  units 

^.^ —      from   15   units  leaves  8  units.     8  tens  from  5  tens 
Diff.  28 

leaves  2  tens. 

The  difference  is  28. 
In  practice,  say  7  from  15  leaves  8;  3  from  5  leaves  2. 

76.  Exercise. 


Copy  and  subtract  : 

/.  37 

5.  56 

9.   70 

13.   27 

19 

89 

18 

18 

2.  42 

6.   74 

10.   62 

14.   43 

29 

36 

9 

19 

3.   45 

7.  40 

//.  53 

n.  67 

26 

29 

18 

19 

4.  81 

8.   91 

12.   61 

16.   46 

38 

82 

14 

4 

ADDITION  AND  SUBTRACTION.  43 


17.   52 

23.   34 

29. 

70 

35.   436 

28 

15 

27 

187 

18.   70 

24.   90 

30. 

52 

36.   491 

36 

44 

36 

238 

19.   32 

25.  21 

31. 

186 

37.   572 

27 

9 

28 

387 

20.   50 

2^.  41 

32. 

192 

38.   500 

41 

29 

49 

283 

21.   93 

27.  56 

33. 

228 

39.   406 

84 

17 

182 

308 

22.  77 

28.   92 

34. 

374 

40.   702 

58 

13 

182 

184 

76.  Exercise. 

Find  the  missing  numbers  : 

/.       8+9+6:rro  3.  2  +         3+         4  = 

16  +  32  4-  27  =  o         20  +  30  4-  40  = 
24  +  41  +  44  =  .        200  +  300  +  400  = 


+     •  = 


2  4-   3  4-   4 


4.  Add  down  ;  subtract  across. 

2.  11  4-  27  +  38  =  .  .  186  -    27  =  • 

9  4-  18  4-  60  =  .  220  -  185  =   . 

36  4-    7  4-    8  =^  407  -  200  =^ 

•  4~     •4"    •  -—   •  •     —    •     — —   • 


44   WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 

77.  Exercise. 

/.  There  are  30  days  in  June,  31  days  in  July,  and  31 
days  in  August.  How  many  days  are  there  in  these  3 
months  ? 

30  is  the  number  of  days  in   June. 

31  "    "  *'         "      "     "    July. 
31   '*    "  **         "      ''     ''    August. 

T  "     ''  **         *'       "     "    the  3  months. 

2.  A  drover  sold  in  one  day  36  cows,  42  hogs,  and  21 
sheep ;  how  many  head  of  stock  did  he  sell  ? 

5.  A  woman  sold  a  dealer  28  pounds  of  poultry,  16 
pounds  of  butter,  and  36  pounds  of  lard  ;  how  many 
pounds  of  produce  did  she  sell  ? 

4.  A  laborer  worked  26  days  in  January,  23  days  in 
February,  and  25  days  in  March  ;  how  many  days  did  he 
work  during  these  months  ? 

5.  I  spent  at  a  grocer's  to-day  36  cents  for  coffee,  24  cents 
for  sugar,  and  78  cents  for  meat ;  how  much  did  I  spend 
in  all  ? 

Note.     Ct.  stands  for  cent  or  cents. 

36  ct.  was  the  amount  spent  for  coffee. 
24  ct.     "      '*         '*  "       "    sugar. 

78  ct.     "       ''         "  "       "    meat. 

?  ''       ''         ''  "       "    all. 

e.  A  clerk  had  $25  of  liis  month's  salary  left  nftor  pay- 
ing a  $20  store  bill  and  $16  for  house  rent ;  wliul  was  liis 
month's  salary  ? 


ADDITION  AND  SUBTRACTION.  45 

7.  After  28  yards  were  sold  from  a  piece  of  bunting, 
there  remained  32  yards  ;  how  many  yards  were  there  in 
the  piece  at  first  ? 

8.  A  farmer  had  36  bushels  of  wheat  and  used  19 
bushels  of  it  for  seed  ;  how  many  bushels  had  he  left  ? 

36  is  the  number  of  bushels  the  fanner  had. 
19  "     *'         "         "         "         he  used  for  seed. 
T    ''     "         "         "         ''         "  had  left. 

9.  A  girl  invited  40  of  her  friends  to  a  picnic,  but  12 
of  them  failed  to  come  ;  how  many  came  ? 

10.  There  are  88  houses  on  a  certain  street ;  if  28  of 
them  are  on  one  side  of  the  street,  how  many  are  on  the 
other  side  ? 

If.  A  huckster  had  80  shad  ;  if  he  sold  all  of  them  but 
18,  how  many  did  he  sell  ? 

12.  A  field  of  68  acres  was  divided  into  two  parts  ;  if 
the  first  part  has  in  it  49  acres,  what  is  the  size  of  the 
second  part  ? 

68  acres  is  the  size  of  tlie  field  divided, 

49      "      "     "      "     "     "     first  part. 
?      "      ''     "      "     "     "      second  part. 

13.  How  much  older  is  my  father  than  I  am,  if  his  age 
is  70  years  and  mine  24  years  ? 

14.  I  sold  32  yards  from  a  piece  of  cheese  cloth  contain- 
ing 60  yards  ;  how  much  remained  ? 

15.  A  dealer  bought  3  pieces  of  printed  lawn  containing 
50  yards,  15  yards,  and  36  yards  ;  how  much  did  he  buy 
in  all  ? 


46    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 

16.  K  flagpole  86  feet  long  had  37  feet  broken  from  the 
top  ;  how  long  was  the  piece  that  was  left  standing  ? 

17.  A.  contractor  has  employed  2G  Italians,  14  Hun- 
garians, and  22  Polanders ;  how  many  laborers  has  he 
employed  ? 

18.  A  boy  had  18  papers  left  after  selling  24  morning 
papers  and  21  evening  papers  ;  how  many  had  he  to 
begin  with  ? 

19.  52  girls  were  given  a  problem ;  if  19  of  them  had 
the  wrong  answer,  how*  many  solved  the  problem  cor- 
rectly ? 

20.  A  drover  sold  a  horse  for  95  dollars  ;  after  he  had 
paid  a  debt  of  27  dollars  out  of  this  sum,  how  much  had 
he  left  ?  I 

21.  K  boy  weighed  101  pounds  and  has  since  gained  19 
pounds  ;  how  much  does  he  now  weigh  ? 

22.  There  were  28  men,  7G  women,  62  boys  and  78 
girls  at  a  Sunday-school  picnic  ;  how  many  persons  were 
there  ? 

23.  There  are  68  apple  trees  in  an  orchard  and  18  more 
pear  trees  than  apple  trees  ;  how  many  trees  are  there 
of  both  kinds  ? 

24.  A  man's  wages  are  42  dollars  a  month  and  his  son's 
18  dollars  less  ;  how  mucli  do  both  earn  in  a  month  ? 

25.  A  drover  had  264  sheep  and  bought  187  more  ;  how 
many  had  he  then  ? 

26.  Find  the  sum  of  three  hundred  fifty,  two  hundred   ' 
twenty-five,  and  one  hundred  eighty. 


MULTIPLICATION,  DIVISION,  MENSURATION.      47 

27.  A  has  275  dollars  and  B  189  dollars  ;  how  much 
would  each  have  if  A  were  to  give  B  186  dollars  ? 

28.  I  owed  0  278  dollars,  but  paid  him  129  dollars  ; 
how  much  do  I  still  owe  him  ? 

29.  A  miller  had  736  bushels  of  wheat  but  sold  478 
bushels  of  it,  and  later  bought  127  bushels ;  how  many 
bushels  had  he  then  ? 

30.  Add  six  hundred  eighty-nine  and  two  hundred  six- 
teen, and  take  four  hundred  four  from  the  sum. 

Multiplication,    Division,   and    Mensuration. 

78.  Oral  ExercisCo 

Count  by  2's  : 

/.  From  2  to  20  ;  from  20  to  2. 
Thus,  2,  4,  6,  etc. ;  then  20,  18,  16,  etc. 
2.  From  1  to  21  ;  from  21  to  1. 

Count  by  3's  : 

5.  From  3  to  30  ;  from  30  to  3. 

4.  From  1  to  31  ;  from  31  to  1. 

5.  From  2  to  32  ;  from  32  to  2. 

79.  Oral  Exercise. 

How  many  are  1  and  1  ? 

1  and  1  are  how  many  times  1?  #+•  =  •• 

How  many  are  2  times  1  ? 

What  are  the  two  parts  of  2  ? 


48    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 

What  are  the  two  parts  of  3?  •#  +  •  =  ••• 

llow  many  are  2  and  2  ? 

2  and  2  are  how  many  times  2?  ••-{-•#  =  ###• 

JIow  many  are  2  times  2  ? 

What  are  the  two  parts  of  4  here  shown  ? 

Are  these  parts  alike  ? 

When  the  parts  of  a  number  are  alike  they  are  said  to 
he  equal. 

Wliat  is  one  of  the  two  equal  parts  of  2?     ••  =  #-4-# 

How  many  ones  are  there  in  2  ? 

What  is  one  of  the  two  equal        ••••  =  ##-f-## 

parts  of  4  ? 
How  many   twos  (2's)  are  there  in  4  ? 
How  many  are  2  times  3  ? 

•  •  • 
What  is  one  of  the  two  equal  parts  of  6  ?  •  •  • 

Each  of  the  ttvo  equal  parts  of  a  7iumber  is  called  one 
half  of  it. 

One  half  is  written  •}. 

What  is  one  half  of  6  ? 

llow  many  3's  are  there  in  6  ? 

llow  many  are  2  times  4  ?  •    •    •    • 

AVhat  is  one  half  of  8  ?•  •    •    •    # 

How  many  4's  are  there  in  8  ? 

How  many  are  2  times  5  ?  •    •    •    •    • 

What  is  i  of  10  ?  •    #    •    •    • 

How  many  5's  are  there  in  10  ? 


MULTIPLICATION,  DIVISION,  MENSURATION.    49 

;Io\v  many  are  2  times  6?  •••••• 

What  is  i  of  12  ?  •••••• 

g_     How  many  6^s  are  there  in  12  ? 

How  many  are  2  times  7?  ••••••• 

What  is  ^  of  14  ?  ••••••• 

How  many  7's  are  there  in  14  ? 

How  many  are  2  times  8?  •••••••• 

What  is  i  of  16?  •••••••• 

How  many  8's  are  there  is  10  ? 

How  many  are  2  times  9?        ••••••••• 

What  is  i  of  18?  ••••••••• 

,    How  many  9's  are  there  in  18  ? 

80.  Oral  Exercise. 

How  many  are  3  times  1  .'' 
i    What  is  one  of  the  tliree  equal  parts  of  3  ? 

Bach  of  the  three  equal  parts   of  a  fiumber  is  called 
I    one  third  of  it. 

One  third  is  written  ■^. 

[    What  is  i  of  3  ? 

How  many  I's  are  there  in  3  ? 

How  many  are  3  times  2  ?  •  • 

What  is  i  of  6  ?  J  J 

How  many  2's  are  there  in  6  ? 
4 


60    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 


IIow  many  are  3  times  3  ?  •  •  • 

What  is  i  of  0  ?  •  •  • 

How  many  3's  are  there  in  9  ? 


How  many  are  3  times  4  ?  •  •  •  • 

What  is  i  of  12  ?  *  •  •  •  • 

#  •  •  • 
How  many  4's  are  there  in  12  ? 


How  many  are  3  times  5  ? 

What  is  i  of  15  ? 

How  many  5's  are  there  in  15  ? 

How  many  are  3  times  G  ? 

What  is  i  of  18  ? 

How  many  G's  are  there  in  18  ? 

How  many  are  3  times  7  ? 

What  is  ^  of  21  ? 

How  many  7's  are  tliere  in  21  ? 

How  many  are  3  times  8  ? 

What  is  i  of  24  ? 

How  many  8's  are  there  in  24  ? 

How  many  are  3  times  9  ? 

What  is  4  of  27  ? 

How  many  9'8  are  there  in  27  ? 


MULTIPLICATION,  DIVISION,  MENSURATION.     51 


81.  The  sign  x  is  read  times. 

82.  Exercise. 

Copy,  supplying  the  missing  numbers  : 


1  +  1  = 
2x1  = 

2  +  2  = 
2x2  = 

3  +  3  = 
2x3  = 


4  +  4  = 
2x4  = 

5  +  5  = 
2x5  = 
G  +  G  - 
2  x  G  = 


7  +  7 
2x7 

8  +  8 
2x8 

9  +  9 
2x9 


83.  Exercise. 

Copy,  supplying  the  missing  numbers,  and  learn 


2x1  = 
2x2  = 
2x3  = 


TABLE. 

2x4  = 
2x5  = 
2  X  G  = 


2x7  = 
2x8  = 
2x9  = 


84.  Exercise. 

Copy,  supplying  the  missing  numbers 


2x1+1= 

2x4  +  4  = 

2x7  +  7=  . 

3x1  = 

3x4=. 

3x7=. 

2x2+2= 

2x5  +  5=  « 

2x8  +  8=  . 

3x2  = 

3x5  = 

3x8=. 

2x3  +  3=  . 

2  X  G  +  6  =  . 

2x9+9=. 

3x3=. 

3x6=. 

3x9=. 

62    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 

85.  Exercise. 

Copy,  supplying  the  missing  numbers,  and  learn  : 

TABLE. 

3x1=.  3x4=.  3x7=. 

3x2=.  3x5=.  3x8=. 

3x3=.  3x6=.  3x9=. 

86.  Exercise. 

Copy,  supplying  the  missing  numbers  : 


/. 

2 

X 

2=  . 

19. 

.  X 

1  =  2 

37. 

3 

X 

.  =9 

2. 

3 

X 

8=  . 

20. 

2  X 

.=  10 

38. 

3 

X 

.  =27 

3. 

2 

X 

7=  . 

21. 

3  X 

7=  . 

39. 

• 

X 

4=  12 

4. 

3 

X 

3=  . 

22. 

.  X 

1  =  3 

40. 

• 

X 

2=4 

6. 

2 

X 

.=  6 

23. 

2  X 

1  =  . 

41. 

3 

X 

.  =3 

6. 

2 

X 

.  =  4 

24. 

3  X 

.=  21 

42. 

• 

X 

6=  18 

7. 

. 

X 

9  =  27 

25. 

2  X 

4=  . 

43. 

2 

X 

.  =  IG 

8. 

3 

X 

5  =  . 

26. 

.  X 

9  =  18 

44. 

• 

X 

3  =  9 

9. 

• 

X 

8  =  24 

27. 

2  X 

3=  . 

45. 

3 

X 

.  =  15 

10. 

• 

X 

2  =  4 

28. 

2  X 

6  =  . 

46. 

. 

X 

7  =  21 

11. 

3 

X 

6  =  . 

29. 

•  X 

5  =  10 

47. 

• 

X 

5  =  15 

12. 

2 

X 

.=  8 

30. 

2  X 

.=  14 

48. 

3 

X 

2=  . 

13. 

2 

X 

5  =  . 

31. 

3  X 

.=  24 

49. 

2 

X 

9=  . 

14. 

2 

X 

.=  2 

32. 

2  X 

8=  . 

50. 

3 

X 

•  =  12 

15. 

2 

X 

.=  18 

33. 

3  X 

9=  . 

51. 

• 

X 

7  =  14 

16. 

2 

X 

.=  12 

34. 

.  X 

1  =  2 

62. 

3 

X 

1  =  . 

17. 

3 

X 

.=  6 

35. 

.   X 

G  =  12 

53. 

3 

X 

.  =  18 

18. 

3 

X 

4=  . 

36. 

•  X 

2  =  6 

64. 

• 

X 

8  =  16 

MULTIPLICATION,  DIVISION,  MENSURATION.     63 

87.  Oral  Exercise. 

/.   What  do  2  ])osUil  cards  cost  ? 

2.  What  do  2  two-cent  stamps  cost  ? 


1  pint. 


1  quart. 


There  are  tivo pints  (pt.)  m  a  quart  (qt.). 

3.  How  many  pints  are  there  in  2  qnarts  ? 

4.  How  many  pints  are  there  in  3  quarts  ? 

5.  What  do  3  one-cent  stamps  cost  ? 

6.  How  many  feet  have  2  horses  ? 

7.  How  many  horses  are  needed  to  make  2  six-horse 
teams  ? 

8r  How  many  horseshoes  will  it  take  to  shoe  3  horses 
all  around  ? 

Measure  a  yard-stick  with  a  foot-rule  to  find  that  : 

There  are  three  feet  (ft.)  in  a  yard  {yd.). 

9.  How  many  feet  are  there  in  2  yards  ? 
70.  How  much  fare  will  two  persons  pay,  if  each  pays 
5  cents  ? 

//.  How  many  days  are  there  in  two  weeks  ? 

12.  How  many  feet  are  there  in  3  yards  ? 

13.  How  many  days  are  there  in  3  weeks  ? 


54   WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 


1  quart. 


1  gallon. 


There  are  four  quarts  in  a  gallon  {gal.), 

14.  How  many  quarts  are  there  in  2  gallons  ? 

15.  How  many  trees  are  there  in  two  rows,  each  con- 
taining 8  trees  ? 

16.  How  many  quarts  are  there  in  3  gallons  ? 

17.  Three  persons  each  pay  5  cents  car  fare  ;  how  much 
do  they  all  pay  together  ? 

18.  How  many  panes  of  glass  are  needed  for  3  windows, 
if  8  panes  are  needed  for  each  window  ? 

19.  I  can  hold  out  9  pounds  in  each  hand ;  what  weight 
can  I  hold  out  with  both  hands  ? 

20.  How  many  persons  are  there  at  3  tables,  if  6  per- 
sons are  at  each  table  ? 

21.  A  grass-plot  is  enclosed  by  3  rows  of  hedge,  each  9 
yards  long ;  how  many  yards  of  hedge  are  there  in  all  ? 


Find  the  cost  of  : 

22.  2  tablets  at  6  cents  each. 

23.  3  lemons  at  3  cents  each. 


MULTIPLICATION,  DIVISION,  MENSURATION.     55 


24.  3  pounds  of  sugar  at  6  cents  a  pound. 

25.  2  sheep  at  4  dollars  apiece. 

88.  Exercise. 

Copy,  writing  in  each  case  the  proper  figure   in  place 
of  X  : 


/.  i  of    2  =  X 

7.  1  of  14  =  X 

13. 

1  of  12  =  X 

2.    i  of     4:  =  X 

8.  ^  of  16  =  X 

14. 

1  of  15  =3  X 

5.  i  of    6  =  X 

9.  i  of  18  ==  X 

15. 

i  of  18  =:  X 

4.  ^  of    8  =  X 

10.  ^ot    3  =  X 

16. 

i  of  21  =  X 

5.  1  of  10  =  X 

11.  ^  of    6  =  X 

17. 

J  of  24  =  X 

6.  i  of  12  =  X 

/2.  1  of    9  =  X 

18. 

i  of  27  =  X 

89.  Oral  Exercise. 

Read  Exercise  88,  supplying  the  missing  numbers. 

90.  Exercise. 

Copy,  supplying  the  missing  numbers  : 

13.  15  =  — fives. 

14.  27  =  — nines. 

15.  14  =  — sevens. 

16.  18  =  —  sixes. 

17.  16  =  —  eights. 

18.  24:=:—  eights. 

91.  Oral  Exercise. 

Read  Exercise  90,  supplying  the  missing  numbers. 

92.  Exercise. 

/.  i  of  8  pints  are  —  pints,  or  —  quarts. 

2.  J  of  a  dozen  are  how  many  ? 

3.  Divide  6  cents  equally  between  2  boys.     What  part 


/. 

2  =  — ones. 

7. 

9  =  —  threes. 

2. 

4  =  —  twos. 

8. 

12  —  — fours. 

3. 

3  =  — ones. 

9. 

12  =:  —  sixes. 

4. 

6  =  —  threes. 

10. 

8  =  — fours. 

6. 

6  =  —  twos. 

11. 

21  =  — sevens. 

6. 

10  =  —  fives. 

12. 

18  =  — nines. 

66    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 

of  the  money  does  each  receive  ?    How  much  does  each 
receive  ? 

4.  If  2  quarts  of  syrup  cost  12  cents,  1  quart  will  cost  ^ 
of  —  cents,  or  —  cents. 

5.  How  much  oil  does  a  stove  burn  in  1  hour,  if  it 
burns  4  quarts  iu  2  hours  ? 

6.  If  I  buy  3  pints  of  ice  cream  for  3  persons,  how  mucli 
will  that  allow  for  each  person  ? 

7.  When  6  quarts  of  milk  fill  3  cans  of  the  same  size, 
how  much  does  each  can  hold  ? 

8.  18  pails  were  tied  in  2  bundles  of  the  same  size  ;  how 
many  were  put  in  a  bundle  ? 

9.  I  wish  to  have  18  pounds  of  sugar  put  up  in  3  pack- 
ages of  the  same  weight  ;  how  much  must  be  put  in  each 
package  ? 

10.  24  children  are  formed  into  3  classes  of  the  same 
size ;  how  many  are  there  in  a  class  ? 

//.  Make  and  answer  a  question  about  J  of  18  cents. 

12.  Mary  spent  15  cents  for  3  pounds  of  sugar.  Ask  a 
question  about  this,  and  give  the  answer. 

13.  A  boy  had  12  cents.  He  spent  ^  of  it  for  fire- 
crackers and  J  of  it  for  ice  cream  ;  how  much  did  he 
spend  for  both  ? 

14.  A  girl  who  earns  9  dollars  and  spends  \  of  it  for 
board  has  how  much  left  ? 

93.  Exercise. 

/.  If  one  lemon  costs  3  cents,  for  G  cents  —  lemons  can 
be  bought. 


MULTIPLICATION,  DIVISION,  MENSURATION.     57 

2.  How  many  3-foot  ropes  can  be  cut  from  a  rope  9  feet 
long  ? 

3.  How  many  times  can  a  gallon  measure  be  filled  from 
12  quarts  of  milk  ? 

4.  How  many  4-horse  teams  can  be  formed  from  8  horses  ? 

5.  How  many  8-foot  ropes  can  be  cut  from  a  rope  16 
feet  long  ? 

6.  How  many  packages  of  oatmeal,  each  containing  4 
pounds,  must  I  buy  to  have  12  pounds  ? 

7.  2  quarts  of  raspberries  at  8  cents  a  quart,  with  5 
cents,  will  pay  for  —  sacks  of  salt  at  3  cents  a  sack. 

8.  Last  week  I  worked  3  days  and  was  paid  6  dollars  ; 
how  much  was  that  a  day  ? 

9.  In  12  pints  there  are  —  quarts. 

10.  15  cents  will  pay  for  —  car  rides  at  5  cents  each. 
//.  How  many  tables  must  be  set  for  24  persons,  if  8 

persons  can  be  seated  at  a  table  ? 

12.  How  many  are  there  in  half  a  dozen  ? 

13.  How  many  are  there  in  one  third  of  a  dozen  ? 

14.  2  pecks  of  potatoes  at  8  cents  a  peck  are  worth  — 
pounds  of  flour  at  4  cents  a  pound. 

15.  If  I  paid  10  cents  for  pencils  at  5  cents    apiece, 
how  many  did  I  buy  ? 

94.  Oral  Exercise. 

Count  by  4's  : 

/.  From  4  to  40.  4.  From  2  to  42. 

2.  From  40  to  4.  5.  From  3  to  43. 

3.  From  1  to  41.  6.  From  43  to  3. 


58    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 


Count  by  5's  : 

7.  From  5  to  50. 

8.  From  50  to  5. 

9.  From  1  to  51. 


10.  From  2  to  52. 
//.  From  3  to  53. 
12.  From  4  to  54. 


95.  Exercise. 

Copy,  supplying  the  missing  numbers  : 


3x1  +  1.= 

3x4+4= 

3x7  +  7=  . 

4x1=. 

4x4  = 

4x7=. 

3x2  +  2=.  . 

3x5+5= 

3x8  +  8=  . 

4x2=. 

4x5  = 

4x8=. 

3x3  +  3=  . 

3x6  +  6=  . 

3x9  +  0=  . 

4x3=. 

4x6  = 

4x9=. 

96.  Exercise. 

Copy,  supplying  the  missing  numbers  and  learn  : 

TABLE. 

4x1=.                    4x4=.  4x7 

4x2=.                    4x5=.  4x8 

4x3=.                    4x6=.  4x9 

97.  Exercise. 

Copy,  supplying  the  missing  num^bers  : 

4x1+1=.            4x4+4=.  4x7+7 

5x1=.        5x4=.  5x7 

4x2+2=.     4x5+5=.  4x8+8 

5x2=.        6x5=.  5x8 

4x3+3=.     4x6+6=.  4x9+9 

5x3=.        5x6=.  5x9 


MULTIPLICATION,  DIVISION,  MENSURATION.     59 

98.  Exercise. 

Copy^  supplying  the  missing  numbers  and  learn  : 

TABLE.  ^ 

5xl=»  5x4=»  5x7=» 

5x2=-  5x5=»  5x8=- 

5x3=-  5x6=«  6x9=- 

99.  Exercise. 

Copy,  supplying  the  missing  numbers  : 


/. 

4  X 

1  =  . 

19. 

5  X  .  =  15 

37. 

4 

X 

.=  8 

2. 

4  X 

.  =  12 

20. 

.  X  5  =  25 

38. 

. 

X 

4=  16 

3. 

4  X 

6=  . 

21. 

4x2=. 

39. 

5 

X 

4=  . 

4. 

4  X 

.=  32 

22. 

.  X  6  =  24 

40. 

5 

X 

5  =  . 

5. 

5  X 

'=  20 

23. 

4  X  5  =  • 

41. 

. 

X 

2  =  8 

6. 

4  X 

4=  . 

24. 

.  X  1  =5 

42. 

5 

X 

.=  25 

7. 

5  X 

.  =  35 

25. 

.  X  7  =  28 

43. 

• 

X 

5  =  20 

8. 

5  X 

7  =  . 

26. 

5  X  .  =  30 

44. 

4 

X 

7=  . 

9. 

•  X 

2  =  10 

27. 

4  X  .  =  20 

45. 

4 

X 

.=  36 

10. 

5  X 

3=  . 

28. 

.  X  6  =  30 

46. 

4 

X 

9  =  . 

11. 

4  X 

.  =  4 

29. 

5x0=. 

47. 

• 

X 

9  =  45 

12. 

•  X 

9  =  36 

30. 

.  X  3  =  15 

48. 

5 

X 

.=  45 

13. 

•  X 

8  =  40 

31. 

4x3=. 

49. 

5 

X 

6=  . 

14. 

4  X 

.  =28 

32. 

5  X  .  =  10 

50. 

4 

X 

.=  16 

15. 

•  X 

7  =  35 

33. 

.  X  1  =  4 

51. 

• 

X 

4  =  20 

16. 

4  X 

8=  . 

34. 

5  X  .  =  5 

52. 

4 

X 

.  =  24 

17. 

5  X 

.  =  40 

35. 

5x8=. 

53. 

5 

X 

2=  . 

18. 

5  X 

1=  . 

36. 

.  X  3  =  12 

54. 

• 

X 

8  =  32 

60    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 

iOO.  Exercise. 

/.  In  one  week  there  are  —  days,  and  in  4  weeks  there 
are  4  times  —  days,  or  —  days. 

2.  ^n  one  yard  there  are  —  feet,  and  in  5  yards  there 
are  5  times  —  feet,  or  —  feet. 

5.  In  one  gallon  there  are  —  quarts,  and  in  4  gallons 
there  are  4  times  —  quarts,  or  —  quarts. 

4.  In  one  quart  there  are  —  pints,  and  in  5  quarts  there 
are  5  times  —  pints,  or  —  pints. 


Q 


1  quart.  1  peck.  1  bushel. 

There  are  8  quarts  (qt.)  in  a  peck  (pk.). 
There  are  Jf.  jjecks  in  a  hnshel  [by.). 

5.  How  many  quarts  are  there  in  4  pecks  ?  in  3  pecks  ? 

6.  How  many  quarts  are  there  in  2  pk.  ?  in  6  pk.  ? 

7.  How  many  pecks  are  there  in  4  bushels  ?  in  5  bushels? 

8.  How  many  pecks  are  there  in  2  bu.  ?  in  3  bu.  ? 
5.  4  X  2  and  how  many  make  a  dozen  ? 

10.  A  dime  and  how  many  cents  will  pay  for  3  pounds 
of  sugar  at  6  cents  a  pound  ? 

//.  How  much  change  should  a  lady  receive  who 
bought  3  pounds  of  soap  at  4  cents  a  pound  and  gave  the 
clerk  a  dime  and  a  5-cent  piece  ? 


MULTIPLICATION,  DIVISION,  MENSURATION.      61 

Find  the  cost  of  : 

12.  2  pencils  at  5  cents  each. 

13.  4  pounds  of  sugar  at  5  cents  each. 

14.  5  oranges  at  2  cents  each. 

15.  4  eye  shades  at  8  cents  each. 

16.  4:  yards  of  ribbon  at  6  cents  a  yard. 

17.  6  packs  of  firecrackers  at  5  cents  a  pack. 

18.  5  loaves  of  bread  at  6  cents  a  loaf. 

19.  4  quarts  of  milk  at  7  cents  a  quart. 

20.  5  papers  of  pins  at  5  cents  a  paper. 

101.  Each  of  the  four  equal  jmrts  of  a  number  is 
called  one  fourth  (i)  of  it. 

102.  Each  of  the  five  equal  parts  of  a  number  is 
called  one  fifth  (^)  of  it. 

103.  Exercise. 

Copy,  writing  in  each  case  the  proper  figure  in  place 
of  X  : 

/.  i  of    4  =  X  7.  i  of  28  =  X  /3.  J  of  20  --  X 

2.  J  of    8  =  X  S.  i  of  32  =  X  /4.  ^  of  25  =r  X 

5.  i  of  12  =  X  5.  i  of  36  =  X  15.  ^  of  30  =  x 

4.  i  of  16  =  X  /O.  i  of    5  =  X  16.  i  of  35  =  X 

5.  \  of  20  =  X  //.  i  of  10  ==  X  17.  ^  of  40  =  X 

6.  i  of  24  =  X  12.  ^  of  15  ==  X  18.  ^  of  45  =  x 

104.  Oral  Exercise. 

Bead  Exercise  103,  supplying  the  missing  numbers. 


62   WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 

105.  Exercise. 

Copy,  supplying  the  missing  numbers  : 

1.  4  —  — ones.      7.  24  =  — sixes.       13.  30  =  — sixes. 

2.  10  =  —  twos.      8.  30  =  —  nines.      14.     6  =  —  ones. 

3.  8  =  — twos.      5.  20  ==  —  fives.        /5.  45  1=  —  nines. 

4.  15  -  —fives.     10.  12  =  —threes.    16.  32  =  —eights. 

5.  16  =  — fours.    //.  25= — fives.       17.  35  =  — sevens. 

6.  20  =  —  fours.   12.  28  =  —  sevens.    18.  40  =  —  eights. 

106.  Oral  Exercise. 

Head  Exercise  105,supplying  the  missing  numbers. 

107.  Exercise. 

/.  If  I  pay  20  dollars  for  4  weeks'  board,  how  much  is 
that  a  week  ? 

2.  I  wish  to  cut  40  yards  of  carpet  into  6  equal  strips  ; 
how  long  must  I  make  each  strip  ? 

3.  10  pigeons  will  make  —  pair  of  pigeons. 

4.  How  many  quart  bottles  can  be  filled  from  8  pints  ? 

5.  How  many  5-cent  boxes  of  indigo  can  be  bought  for 
20  cents  ? 

6.  A  school  of  24  boys  was  divided  into  4  equal  classes ; 
how  many  boys  does  each  class  contain  ? 

7.  How  many  are  24  less  8  divided  by  4  ? 

8.  How  many  times  must  I  reach  into  a  box  containing 
eggs  to  count  out  2  dozen,  if  I  take  out  6  each  time  ? 

9.  In  10  quarts  there  are  —  pocks. 

10.  How  many  pieces,  each  3  feet  long,  can  I  cut  from 
15  feet  of  twine  ? 


MULTIPLICATION,  DIVISION,  MENSURATION.      63 

11.  K  painter  received  18  dollars  for  painting  a  barn  ; 
if  he  worked  6  days,  how  much  did  he  earn  each  day  ? 

12.  IIow  many  5-cent  combs  can  be  bought  for  40  cents  ? 

13.  How  many  gallon  jugs  can  be  filled  with  20  quarts 
of  vinegar  ? 

14.  Allowing  a  pound  of  meat  for  2  persons,  how  much 
meat  should  be  bought  for  a  dinner  at  which  10  persons 
are  to  be  served  ? 

15.  How  many  pounds  of  sugar  at  4  cents  a  pound  can  be 
bought  in  exchange  for  3  pounds  of  lard  at  8  cents  a  pound  ? 

20  is  called  a  score. 

16.  How  many  are  there  in  i  of  a  dozen  ?  In  i  of  a 
score  ? 

108.  One  of  the  two 
equal  parts  of  1  is  i  ;  or 
ioflisi 

One  of  the  three  equal 
parts  of  1  is  i  ;  or  ^  of  1 
is  i. 

J  of  2  is  i  plus  -J,  or  f, 
read  two  thirds. 

109.  Find  J  of  5. 

Answer  thus  :  |  of  5  is  2  and  1  over  ;  i  of  1  is  |. 
The  answer  is  two  and  one  half  ;  written  2|. 

110.  Oral  Exercise. 

Find: 
/.  J  of  3.  3.  J  of    9.  5.  I  of  11.  7.  i  of  17. 

2.  J  of  7.  4.  i  of  13.  6.  ^  of  15.  8.  J  of  19. 


64   WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 


111.  Find  i  of  17. 

Answer  thus  :  i  of  17  is  5,  and  2  over  ;  J  of  2  is  |. 
Tlie  answer  is  5J. 

112.  Oral  Exercise. 
Find  : 

/.  i  of    5.  5.  4  of    4.  9.  J  of  10.  13.  i  of  13. 

2.  J  of    8.  6.  I  of    7.  10.  i  of  22.  14.  J  of  25. 

3.  \  of  11.  7.  i  of  IG.  //.  i  of  23.  15.  J  of  26. 

4.  J  of  14.  8.  i  of  19.  /2.  i  of  28.  /ff.  J  of  29. 

113.  Exercise. 

/.  J  of  3  pints  is  —  pints. 

2.  Find  J  of  7  feet. 

5.  Divide  8  yards  of  ribbon  equally  among  3  girls. 

4.  If  a  man  earned  3  dollars  in  2  days,  in  1  day  he 
earned  J  of  —  dollars,  or  —  dollars. 

5.  A  lady  bought  3  quarts  of  plums  for  25  cents ;  how 
much  were  they  a  quart  ? 

6.  At  7  cents  a  quart,  what  will  a  pint  of  milk  cost  ? 

7.  An  iron   rod  9  feet  long  was  divided  into  2  equal 
parts  ;  find  the  length  of  each  part. 

8.  When  3  loaves  of  bread  cost  10  cents,  how  much 
does  one  loaf  cost  ? 

9.  A  family  that  uses  a  pound  of  tea  in  3  weeks,  uses 
how  much  per  week  ? 

10.  When  3  quarts  of  oysters  fill  2  cans  of  equal  size, 
how  much  does  each  can  hold  ? 

//.  How  many  days  are  there  in  lialf  a  week  ? 


MULTIPLICATION,  DIVISION,  MENSURATION.     65 
114.  How  many  2's  in  7  ? 

3^  twos 


(2)      (2)      (2)    (lof2) 


How  many  3's  in  11  ? 


33-threes 


11  =  3f  threes. 

(3)        (3)         (3)      (|-of3) 

116.  Exercise. 

Supply  the  missing  numbers,  using  lines  to  illustrate  : 
/.  5  =  — twos.  4.  7  =  — threes.     7.  11  =  — twos. 

2.  4:  =  —  threes.         5.  3  =  —  twos.       8.  10  =  —  threes. 
5.  9  =  — twos.  6.  8  —  —  threes.    9.   14  =  — threes. 

116.  How  many  2's  in  7  ? 

Answer  thus:  There  are  3  twos  in  7  and  1  over. 
This  1  over  is  ^  of  another  2. 
Therefore,  the  number  of  2's  in  7  is  3i,  read 
three  and  one  half. 

117.  Oral  Exercise. 
How  many  : 

/.  2's  in  3  ?      3.  2's  in  13  ?      5.  2's  in  17  ?     7.  2's  in  19  ? 
2.  3's  in  5  ?      4.  3's  in  20  ?      6.  3's  in  16  ?     8.  3's  in  25  ? 
5 


66    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 

118.  Exercise. 

/.  In  7  pints  there  are  —  quarts  —  pint. 

2.  In  11  feet  there  are  —  yards  —  feet. 

3.  How  many  2-cent  stamps  can  be  bought  for  15  cents, 
and  what  change  will  there  be  ? 

4.  17  ducks  will  make  how  many  pair,  and  leave  how 
many  over  ? 

5.  A  liveryman  hired  out  21  horses,  9  of  them  for 
driving  singly  and  the  others  for  double  teams  ;  how 
many  double  teams  were  there  ? 

6.  How  many  sheets  of  drawing  paper  at  3  cents  each 
can  be  bought  for  10  cents,  and  how  much  change  will 
there  be  ? 

7.  If  one  yard  of  braid  costs  2  cents,  for  9  cents  as  many 
yards  can  be  bought  as  —  cents  is  contained  times  in  — 
cents,  or  4^. 

8.  At  2  cents  a  pound,  how  many  pounds  of  putty  can 
be  bought  for  7  cents  ? 

9.  A  man  who  earns  2  dollars  a  day  must  work  how 
many  days  to  earn  11  dollars  ? 

10.  How  many  bags  of  grain  has  a  farmer,  if  he  has  28 
bushels  in  3-bushel  bags  ? 


119.  Exercise. 


/.  What  is  each  of  the  two  equal  parts  of 
1  called  ? 

2.  How  many  halves  are  there  in  1  ? 

3.  Read  :  ^  ;  |. 


MULTIPLICATION,  DIVISION,  MENSURATION.      67 


4.  What  is  each  of  the  four  equal  parts  of 
1  called  ? 

5.  What  are  two  of  the  four  equal  parts  of 
1  called  ? 

6.  Read:  i;f;  };*. 


i  of  2  :=  J  +  J,  or  f ,  or  J. 
7.  Show  that  J  of  3  =  }. 


120.  Find  \  of  14. 

Answer  thus  :  i  of  14  is  3  and  2  over  ;  i  of  2  is  |,  or  \, 
The  answer  is  3i. 

121.  Oral  Exercise. 

Find  : 

/.  \  of  5.  4.  \  of  9.         7.  \  of  25. 

2.  \  of  7.  5.  \  of  11.       8.  \  of  27. 

5.  J  of  6.  ff.  J  of  10.       9.  J  of  22. 


10.  \  of  31. 
//.  }of  33. 
12.  iof34. 


122.  How  many  4's  are  there  in  14  ? 


****     ****     ****  ^**    14  =  3^ fours. 

4  4  4  iof4 

123.  Exercise. 

Supply  the  missing  numbers,  making  drawings  to  illus- 
trate : 

/.  9  =  —  fours. 
^.  7  =  —  fours. 
5.  6  =  —  fours. 


4.  10  =  —  fours. 

5.  13  =  —  fours. 

6.  22  =  —  fours. 


7.  15  =  —  fours. 

8.  23  =  —  fours. 

9.  25  =  —  fours. 


68    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 


124.  Exercise. 

/.  At  4  cents  a  yard,  how  many  yards  of  lace  can  I 
buy  for  a  dime,  and  what  change  will  there  be  ? 

2.  In  19  quarts  there  are  —  gallons  —  quarts. 

3.  At  7  dollars  a  bushel,  what  is  a  peck  of  clover  seed 
worth  ? 

4.  When  a  gallon  of  oil  costs  9  cents,  how  much  is  it  a 
quart  ? 

5.  What  is  each  of  the  five  equal  parts  of  1  called  ? 

6.  Read:  i;  |;|;  t;|. 

7.  Find  ^  of  each  number  in  the  outer  ring  : 


Suggestion,  Solve  as  in  S  120. 


8.  Show  that  there  are  2|  fives  in  12. 

9.  2  pounds  of  lard  at  9  cents  a  pound  will  buy  how 
many  yards  of  lining  at  4  cents  a  yard  ? 

10.  A  boy  had  22  cents  in  5-cent  and  one-cent  pieces. 
He  had  7  one-cent  pieces ;  how  many  5-cent  pieces  had  he? 

//.  How  many  5-ccnt  packages  of  fire-crackers  can  a 
boy  get  for  a  pair  of  pigeons  worth  28  cents  ?  How  much 
money  will  he  have  left? 

12.  A  vessel  that  holds  9  quarts  has  in  it  1  quart  of 


MULTIPLICATION,  DIVISION,  MENSURATION.      69 

water  ;  liow  iiuiiiy  gallons   of  water  can  still   be  ])oured 
into  it  ? 

13.  I  have  four  pieces  of  money,  together  worth  12  cents. 
Two  of  them  are  one -cent  pieces  ;  what  are  the  other 
two  ? 

125.  3  X  23  =  23  +  23  +  23  =  G9.  The  answer  may 
be  found  by  a  shorter  method  by  making  use  of  the  Table 
on  p.  120,  thus  : 

^3  3  X  23  =  3  X  (3  units  +  2  tens). 

_£.  3x3  units  =  9  units,  which  we  write  in  units'  place. 

69  3x2  tens  =  6  tens,  which  we  write  in  tens'  place. 

The  answer  is  69. 

In  practice,  say  3  times  S  are  9  ;  3  times  2  are  6. 

126.  The  process  of  taking  a  number  as  many 
times  as  there  are  units  in  anotlier  number  is  called 
Multiplication.  The  answer  in  multiplication  is  called 
the  Product. 

127.  Multiply  21  by  2. 
Multiplicand         21 
Multiplier  2 
Product                  42 

128.  Exercise. 

Copy  and  multiply,  and  state  which  number  is  the  mul- 
tiplicand, which  the  multiplier,  and  which  the  product : 


70    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 


/.  42 

7. 

30 

13. 

41 

19. 

201 

25. 

203 

2 

3 

5 

4 

3 

2.   33 

8. 

21 

14. 

120 

20. 

212 

2ff. 

121 

3 

3 

2 

4 

4 

3.   20 

9. 

21 

15. 

203 

21. 

200 

27. 

101 

2 

4 

2 

4 

5 

4.   40 

10. 

32 

16. 

101 

22. 

112 

28. 

323 

2 

4 

3 

4 

3 

5.  13 

11. 

40 

17. 

132 

23. 

102 

29. 

213 

3 

4 

3 

4 

3 

6.   32 

12. 

31 

18. 

332 

24. 

111 

30. 

222 

3 

5 

3 

5 

4 

129.  One  of  the  3  equal  parts  of  39  may  be  found  thus : 

i  of  39  =  i  of  (3  tens  +  9  units). 

i  of  3  tens  =  1  ten,  which  we  write  in  tens'  place. 

■^*^         3  of  9  units  =  3  units,  which  we  write  in  units'  place. 
The  answer  is  13. 
In  practice,  my  ^  of  d  is  1  ;  ^  of  d  is  S. 

130.  Finding  one  of  the  equal  parts  of  a  numbei 
is  called  Division.  The  answer  in  division  is  called 
the  Quotient. 

131.  The  sign  of  division  is  -^,  read  divided  by. 

Thus,  24  -s-  3  is  read  ^4  divided  by  3. 
Note.     24-1-3  may  also  be  expressed  3)24. 


MULTIPLICATION,  DIVISION,  MENSURATION.     71 


132.  Divide  28  by  2. 
Divisor       2)28  Dividend 
Quotient  14 

133.  Exercise. 

Copy  and  divide,  and  state  which  number  is  the  divi- 
dend, which  the  divisor,  and  which  the  quotient: 

/.  2)42  7.  3)60  13.  4)80  19.  2)406  25.  4)444 

2.  2)64  8.  3)33  14.  4)40  20.  3)333  26.  4)480 

5.  2)86  5.  3)69  /5.  5)^  21.  3)369  27.  4)408 

4.  2)40  /O.  4)^  /^.   5)55  22.  3)960  25.   5)555 

5.  3)36  //.  4)44  /7.  2)222  23.  3)609  25.  5)505 
5.  3)96  /2.  4)_48  /5    2)264  24.  3)^06  50.   5)550 

134.  Multiply  45  by  4. 

45         4  X  45  =  4  X  (5  units  +  4  tens). 
4         4x5  units  =  20  units,  or  3  tens. 
180         Write  0  in  units'  place,  and  add  2  tens  to  the  next 
product. 

4x4  tens  =  16  tens,  which  added  to  the  2  tens  makes  18  tens, 
or  1  hundred  and  8  tens. 

Write  8  in  tens'   phxce  and  1  in  hundreds'  place. 

The  product  is  180. 

In  practice^  say  If,  times  5  are  20  ;  Jf,  times  Jf,  are  16,  and  2  are  18. 


72    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 


135. 

Exercise. 

Copy  and  multiply  : 

/.  37 

7.  34 

13. 

66 

19.  46 

25. 

192 

2 

3 

4 

5 

5 

2.  39 

5.   87 

14. 

89 

20.  55 

26. 

287 

2 

3 

4 

5 

3 

3.  48 

9.   64 

15. 

94 

2/.   136 

27. 

498 

2 

3 

4 

2 

2 

4.  56 

10.  95 

16. 

19 

22.   176 

28. 

185 

2 

3 

5 

3 

5 

5.  75 

//.  23 

17. 

28 

23.   135 

29. 

225 

2 

4 

_5 

3 

4 

^.  29 

12.  57 

18. 

37 

24.   186 

30. 

199 

_3 

4 

5 

4 

4 

136. 

Exercise. 

/.  How  many  pails  are    there  in  4  bundles,   if    each 
bundle  contains  52  pails  ? 

52  is  the  number  of  pails  in  1  bundle. 
4 

?    is  the  number  of  pails  in  4  bundles. 
2.  Mow  many  are  there  in  4  score  ? 


MULTIPLICATION,  DIVISION,  MENSURATION.    73 

3.  How  many  oranges  are  there  in  3  boxes  each  con- 
taining 36  oranges  ? 

4.  A  bushel  of  oats  weighs  32  pounds  ;  find  the  weight 
of  3  bushels  of  oats. 

82  pounds  is  the  weight  of  1  bushel  of  oats. 
3 
?     pounds  is  the  weight  of  3  bushels  of  oats. 

5.  A  bushel  of  wheat  weighs  60  pounds  ;  find  the  weight 
of  5  bushels  of  wheat. 

6.  How  much  will  a  clerk  save  in  4  months,  if  he  saves 
$26  a  month  ? 

There  are  24  Jiours  {hr.)  in  one  day  {da.). 

7.  How  many  hours  are  there  in  4  days  ? 

8.  A  barrel  of  flour  weighs  196  pounds;  find  the  weight 
of  4  barrels  of  flour. 

9.  Find  the  cost  of  3  pounds  of  butter  at  23  cents  a 
pound. 

10.  Find  the  cost  of  4  pounds  of  ham  at  16  cents  a 
pound. 

//.  K  there  are  24  sheets  of  paper  in  a  box,  how  many 
sheets  are  there  in  4  boxes  of  the  same  size  ? 

12.  Find  the  cost  of  3  hundred  shad  at  $25  per  hundred. 

137.  Divide  312  by  4. 

4  )312        i  of  812  =  i  of  (31  tens  +  2  units). 
78        i  of  31  tens  =  7  tens,  and  8  tens  over. 
3  tens  +  2  units  =  32  units. 
\  of  32  units  =  8  units.     The  quotient  is  78. 
In  practice,  say  i  ^  31  isl  and  8  over  ;  i  ^  82  is  8. 


74    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 


138.  Exercise 

Copy  and  divide  : 

/.  2)34 

/2. 

4)64 

23. 

7)91 

34.  3)252 

2.  2)76 

13. 

4)72 

24. 

8)96 

35.  3)405 

3.  2)92 

14. 

4)76 

25. 

7)98 

36.  4)164 

4.  2)58 

15. 

4)92 

26. 

2)124 

37.  4)876 

5.  2)98 

16. 

5)65 

27. 

2)136 

38.  4)172 

6.  3)42 

17. 

5)75 

28. 

2)154 

39.  4)952 

7.  3)48 

18. 

5)85 

29. 

2)306 

40.  4)708 

8.  3)54 

19. 

5)95 

30. 

2)178 

41.  5)515 

9.  3)72 

20. 

6)72 

31. 

3)153 

42.  5)585 

10.  3)84 

21. 

6)84 

32. 

3)657 

43.  5)195 

//.  4)56 

22. 

6)90 

33. 

3)174 

44.  5)705 

139.  Exercise. 

/.  A  dealer  bought   4  dozen  eggs  for  84  cents  ;  how 
much  was  that  a  dozen  ? 

84  cents  was  the  cost  of  4  dozen. 
4)  84  cents 

?   cents  was  the  cost  of  1  dozen. 

2.  IIow   much   was  paid  a  week   to   a  workman   who 
received  $75  for  5  weeks'  work  ? 


MULTIPLICATION,  DIVISION,  MENSURATION.     75 

3.  A  piece  of  carpet  measuring  72  yards  was  cut  into 
4  strips  of  equal  lengtli ;  find  the  length  of  each  strip. 

4.  56  chairs  are  to  be  arranged  in  4  rows  with  the  same 
number  in  each  row  ;  how  many  must  be  put  in  a  row  ? 

56  is  the  number  to  be  put  in  4  rows. 
4)56 

?  is  the  number  that  must  be  put  in  a  row. 

5.  144  handkerchiefs  were  put  up  in  3  boxes,  the  same 
number  being  in  each  box  ;  how  many  were  put  in  a  box  ? 

6.  5  bushels  of  feed  weighed  225  pounds  ;  what  did  each 
bushel  weigh  ? 

7.  A  farmer  sold  2  cows,  the  first  for  35  dollars  and  the 
second  for  43  dollars  ;  what  was  the  average  selling  price  ? 

35  dollars  was  the  selling  price  of  the  first. 
43       "  "  "  "         ''      second. 

~78       "■  "  "  "         "      both. 

2778       "• 

?    was  the  average  selling  price. 

8.  A  merchant  bought  2  hams  ;  the  first  weighed  23 
pounds  and  the  second  25  pounds  ;  what  was  their  average 
weight  ? 

140.  Finding  hov7  many  times  3  feet  is  contained 
in  12  feet  is  measuring  12  feet  by  3  feet. 

141.  The  process  of  measuring  one  number  by 
another  is  Mensuration. 

142.  The  sign  of  mensuration  is  :,  read  measured 
hy. 

Thus,  12  :  3  is  read  12  measured  by  3, 


76    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 

143.  The  number  of  4'8  in  12  is  3 ;  that  is,  12:4  =  3. 
But  12^4  =  3. 

Therefore,  12:4  =  12^4.     Therefore, 
To  measure  one  number  hy  another^  divide  the 
first  hy  the  second, 

144.  Oral  Exercise. 
Measure  : 

/.  12  by  4.  Answer  thus  :  12  :  4  =  12  -i-  4,  or  3. 

2.  14  by  2.  6.  32  by  4.  10.  28  by  4.        /4.  24  by  4. 

5.  24  by  8.  7.  30  by  5.  //.  40  by  5.        15.  35  by  5. 

4.  18  by  2.  8.  Ti  by  3.  12.  32  by  4.        16.   18  by  3. 

5.  27  by  3.  9.  36  by  4.  13.  21  by  3.        17.  25  by  5. 

145.  2  feet  is  contained  in  14  feet  as  many  times 
as  2  is  contained  in  14  ;  that  is, 

14  feet :  2  feet  =  14  :  2,  or  14  -^  2.     Therefore, 

To  measure  one  amount  hy  another^  divide  the 
number  of  units  in  the  first  by  the  number  of  units 
in  the  second, 

146.  Oral  Exercise. 

Measure  : 

/.  14  feet  by  2  feet. 

Answer  tlms  :   14  feet  :  2  feet  =  14  -*-  2  =  7. 

2.  16  cents  by  2  cents.  7.  45  dollars  by  5  dollars. 

3.  24  yards  by  3  yards.  8.  28  men  by  4  men. 

4.  20  quarts  by  4  quarts.  9.  30  feet  by  5  foot. 

5.  25  days  by  5  days.  10.  20  pounds  by  4  pounds. 
B.  36  quarts  by  4  quarts.  //.  32  gallons  by  4  gallons. 


MULTIPLICATION,  DIVISION,  MENSURATION.     77 

147.  Exercise. 

/.  A  man  bought  144  colored  crayons  in  boxes  contain- 
ing four  pieces  each ;  how  many  boxes  did  he  buy  ? 

144  was  the  number  of  pieces  bought. 
4  was  the  number  of  pieces  in  each  oox. 
4) -144 

?     was  the  number  of  boxes  bought. 

2.  How  many  2-cent  stamps  can  be  bought  for  60 
cents  ? 

3.  How  many  gallon  jugs  can  be  filled  with  64  quarts 
of  vinegar  ? 

A.  How  many  3-bushel  bags  are  needed  to  hold  168 
bushels  of  grain  ? 

5.  How  many  horses  can  be  shod  all  around  with  264 
horseshoes  ? 

6.  How  many  bushels  are  there  in  72  pecks  ? 

7.  How  many  5-cent  pieces  will  make  75  cents  ? 

8.  144  dozen  eggs  are  to  be  packed  in  crates,  each  hold- 
ing 4  dozen  ;  how  many  crates  are  needed  ? 

9.  At  5  cents  a  pound,  how  many  pounds  of  sugar  can 
be  bought  for  90  cents  ? 

10.  How  many  yards  long  is  a  rope  that  is  426  feet 
long  ? 

//.  If  a  family  use  2  quarts  of  milk  a  day,  in  how  many 
days  will  they  have  used  5  gallons  of  milk  ? 

12.  In  how  many  days  will  a  boy  save  60  cents,  if  he 
saves  5  cents  a  day  ? 


78    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 


Forming,   Writing,  and    Reading   Numbers  from 
lOOO  to    lOOOO. 

148.  Ten  hundred  make  one  thousand^  written 
1000. 

Two  thousand  is  written  2000 ;  three  thousand, 
3000. 

One  thousand  one  is  written  1001. 

One  thousand  ten  is  written  1010. 

Five  thousand  two  hundred  is  written  5200. 

Four  thousand  seven  hundred  twenty  is  written 
4720. 

Nine  thousand  nine  hundred  ninety-nine  is 
written  9999. 

149.  Exercise. 

Write  in  figures: 
/.  Three  thousand.  7.  Seven  thousand. 

2.  Four  thousand.  8.  Eight  thousand. 

3.  Five  thousand.  9.  Nine  thousand. 

4.  Six  thousand.  10.  One  thousand  two. 

5.  Two  thousand  twenty.  //.  Six  thousand  five  hundred. 

6.  Four  thousand  forty.     12.  Nine  thousand  fifteen. 

13.  Seven  thousand  six  hundred  eighty-four. 

14.  Nine  thousand  four  hundred  forty. 

15.  Eight  thousand  six  hundred  four. 

16.  Seven  thousand  eight  hundred  sixty-six. 

17.  Nine  thousaiid  eiglit  hundred  twenty. 

18.  Four  tliousand  nine  hundred  ninety-nine. 

19.  One  thousand  one  hundred  one. 


ROMAN  NOTATION.  79 


160.  Exei 

else. 

Read, 

and  write  in  words  : 

/.  400C. 

5.  1200. 

9. 

1009. 

13. 

8777. 

2.   7000. 

6.   5021. 

10. 

9090. 

14. 

G666. 

3.   7002. 

7.  9888. 

11. 

5007. 

15. 

2754. 

4.   1011. 

8.   7232. 

12. 

8201. 

16. 

3215. 

Roman  Notation. 

151.  Capital  letters  are  often  used  instead  of 
figures  in  numbering  chapters  and  sections  of  books, 
pages  of  prefaces  and  introductions,  divisions  on 
clock  dials,  etc. 

This  method  of  writing  numbers  is  called  the 
Roman  System. 

I=:l        Y  =  5        X  =  10        L  =  50        C  =  100 

D  =  500        M  =  1000. 

Note.  The  number  for  which  a  letter  stands  \b  called  the 
value  of  the  letter. 

152.  If  a  letter  is  repeated  its  lvalue  is  repeated. 

Thus,  II  =  2  ;  III  =  3 ;  xx  =  20  ;  cc  =  200  ;  mmm  =  3000. 

153.  If  a  letter  is  written  before  one  of  greater 
7)alue^  the  difference  of  tlielr  values  is  expressed. 

Thus,  IV  =  4  ;  ix  =  9  ;  xl  =  40  ;  xc  =  90. 

154.  If  a  letter  or  a  combination  of  letters  is 
written  after  a  letter  of  greater  value,  the  sum  of 
the  values  is  expressed. 

Thus,  VI  =r  6  ;  xiii  =  13  ;  lv  =  55  ;  lix  =  59. 


80    WHOLE  NUMBERS  AND  FitACTIONAL  PARTS. 

155.  A  dash  placed  over  a  letter  or  a  combina- 
tion of  letters  multiplies  its  value  by  1000. 

Thus,  V  =  5000  ;  vli  =  7000  ;  Tx  =  9000. 

156.  Units,  tens,  hundreds,  and  thousands  only 
are  written  according  to  the  principles  given  above. 


TABLE. 

Units. 

Tens. 

Hundreds. 

Thousands. 

I         =  1 

X 

= 

10 

C 

=  100 

M 

=  1000 

II        =2 

XX 

= 

20 

CC 

=  200 

MM 

=  3000 

III      =  3 

XXX 

= 

80 

CCC 

=  300 

MMM 

=  3000 

IV      =  4 

XL 

= 

40 

CD 

=  400 

IV 

=  4000 

V        =5 

L 

= 

50 

D 

=  500 

V 

=  5000 

yi     =  6 

LX 

= 

60 

DC 

=  600 

vT 

=  COOO 

VII     =  7 

LXX 

= 

70 

DCC 

=  700 

vli 

=  7000 

VIII  =  8 

LXXX 

= 

80 

DCCC 

=  800 

VIII 

=  8000 

IX      =9 

xc 

= 

90 

CM 

=  900 

ix 

-  9000 

157.  In  writing  numbers  between  those  in  the 
above  table,  hundreds  are  written  after  thousands, 
tens  after  hundreds,  and  units  after  tens. 

Thus,  15G7  is  written  mdlxvii  ;  2053,   mmliii  :  4508,  fvDViii  ; 

3600,  MMMDC. 

Note,     hit  =  4  ;  xxxx  =  40  ;  cccc  =  400,  are  also  iu  use. 

158.  Exercise. 

Write  by  the  Roman  method  : 
/.  The  nine  simple  numbers.     3.  The  nine  hundreds. 
2.  The  nine  tens.  4.  The  first  nine  thousands. 

The  numbers  between 

5.  10  and  20.  8.  40  ami  50.  //.  70  and  80. 

6.  20  and  30.  9.  50  and  60.  12.  80  and  90. 

7.  30  and  40.  10.  60  and  70.  13.  1)0  and  100. 


UNITED  STATES  MONEY. 


81 


The  following  numbers  : 

U.  101.                17.   UO. 

20. 

400. 

'23. 

1899. 

15.   120.                 18.   149. 

21. 

6G6. 

24. 

1 90G. 

16.   150.                 19.   194. 

22. 

924. 

25. 

1910. 

Write  in  figures  the  following  : 


26.  IX. 

27.  XIV. 

28.  XIX. 

29.  XXXIV. 

30.  XXXVIII. 

31.  XLIX. 

32.  LXIV. 


33.  XCIX. 

34.  CLX. 
55.  CLXII. 
55.  CCCLXV. 
37.  CDIX. 
55.  CDLXIV. 
39.  DCCCV. 


40.  MCM. 

4/.  MCMV. 

42.  MCMIX. 

43.  MCMXIX. 

44.  MDCCCXCIX. 

45.  VII. 

46.  VCCIV. 


United  States  Money. 

159.  In  the  money  of  the  United  States, 

10  cents  =  1  dime. 
10  dimes  =  1  dollar. 
100  cents  =  1  dollar. 

160.  The  dollar  sign  is  $.  It  is  placed  before  the 
number  of  dollars  to  be  expressed. 

Thus,  5  dollars  is  expressed  $5. 

161.  If  a  sum  of  money  consists  of  dollars  and 
cents,  a  point  called  the  Decimal  Point  is  written 
between  the  number  of  dollars  and  the  number  of 
cents.     If  the  number  of  cents  is  less  than  ten,  a 

6 


82    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 

cipher  is  written  between  the  decimal  point  and  the 
number  of  cents. 

Thus,  S  dollars  25  cents  is  expressed  $3.25,   and  Jf.  dollars  8 
cents  is  expressed  $1^.08. 

162.  The  abbreviation  for  the  word  cent  or  cents 
is  ct.  or  ^. 

Thus,  25  cents  may  be  expressed  25  ct.  or  25i^. 

163.  A  sum  of  money  less  than  a  dollar,  such  as 
36  cents,  may  also  be  expressed  $0.36. 


164.  Oral  Exercise. 

Eead: 

/.  $2.            3.  $15.25. 

5.  $0.62. 

7.  $88.66. 

2.  $3.05.       4.  $37.50. 

6.  $0.07. 

8.  $  0.12^. 

165.  Exercise. 

Write,  using  the  dollar  sign  : 

/.  6  dollars.  4.  7  dollars  5  cents. 

2.  75  dollars  25  cents.      5.  87  cents. 

3.  11  dollars  15  cents.      6.  9  cents. 

166.  Exercise. 

/.  How  many  cents  are  there  in  $3  ?  $5  ?  $12  ? 
2.  Change  $4.36  to  cents. 

$4.36  =  436  ct. 
5.  How  many  cents  are  there  in  $2.25  ?  $4.36  ?  $14.36  ? 


MAKING  CHANGE.  83 

4.  Change  to  cents  : 

$6.84;  $9.32;  $15.36;  $28.45. 

5.  Change  600  ct.  to  dollars. 

600  ct.  =  $6.00,  or  $6. 

6.  Change  to  dollars  : 

500  ct.  ;  900  ct.;  1500  ct.  ;  2800  ct. 

7.  Change  324  ct.  to  dollars  and  cents. 

324  ct.  =$3.24. 

8.  Change  to  dollars  and  cents  : 

539  ct.;  608  ct.;  3125  ct.;  6175  ct. 

Making  Change. 

167.  A  man  bought  a  fork  for  03^/  and  gave  the  store- 
keeper a  dollar  bill ;  how  will  the  storekeeper  make  the 
change,  using  as  few  pieces  of  money  as  possible  ? 

The  storekeeper  will  give  the  man  2  cents,  a  dime,  and 
a  quarter-dollar.  He  may  count  out  the  change,  saying, 
63  ct.,  65  ct.,  75  ct.,  1  dollar. 

168.  Oral  Exercise. 

Make  the  change  in  the  following  examples,  using  as 
few  pieces  of  money  in  each  case  as  possible  ;  and  state 
how  the  change  may  be  counted  out : 

Sales.  Money  Paid. 

/.  17^'.  Quarter-dollar. 

2.  32^.  Half-dollar. 

3.  63r/.  Dollar. 


84    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 

4.  62^'.  Half-dollar  and  quarter-dollar. 

5.  12^'.  A  dime  and  a  5-cent  piece. 

6.  33^'.  A  quarter  dollar  and  a  dime. 

7.  $1.15  ^2-bill. 

8.  $2.12  $2-bill  and  a  $l-bill. 

9.  $3.28  $5-bill. 
10.  9^.  $2-bill. 

//.  $28.37.         $20-bill  and  a  $10-bill. 

Addition  and  Subtraction. 

169.  Add: 

3745  In  practice  we  ma}'  say  : 

2G34  3,  7,  13.     Write  2  in  units'  place  and  add   1  to  the 

1853         tens. 

8232  Then  1,  6,  9,  13.     Write  3  in  tens'  place  and  add 

1  to  the  huncl reels. 

Then  1,  9,  15,  23.  Write  2  in  the  hundreds'  place  and  add  2  to 
the  tliousands. 

Then  2,  3,  5,  8.     Write  8  in  the  thousands'  place. 

The  sum  is  8233. 


170.  Exercise. 

Copy  and  add  : 

/.  345    3.   378 

5.  765 

7.  5482 

9.   3482 

870      745 

080 

2708 

2351 

2.   421    4.   987    6.   2768    8.   5832  10.   7683 
899      876      1289      3168      2297 


ADDITION  AND  SUBTRACTION. 


85 


r/.  3275 

13. 

2845 

15.   3754 

17.   329 

19. 

328 

876 

738 

2832 

896 

798 

49 

897 

1709 

539 

854 

654 
29 

r2.  832 

14. 

2597 

16.   763 

18.   327 

20. 

377 

2375 

3685 

854 

83 

479 

1783 

1763 

963 

785 

901 

78 

637 

507 

Find  the  missing  numbers  by  adding  down  and  across 


21.  234  +  567  +  728  = 

109  +  106  +    34  = 

48  +  729  4-  138  = 


•+•     +     •     = 

22.  728  +  925  +  324  = 

75  +  109  +      7  = 
768  +  325  +  785  =: 


+ 


23.  308  +    78  +  230  = 

106  +      7  +    98  = 

99  +  126  4-      3  = 


+ 


+     •    = 


24.  1024  +  276  +  325 

106  +72+78 
48  +      9  +      3 


+      .    = 


Find  the  missing  numbers  by  multiplying   across  and 
adding  down  ; 

25.  3  X  228  =   .  26.  A:  X  164  =   .  27.  5  x     75  = 

3  X  456  =   .  4  X  108  =   .  5  x  110  =^ 

3  X  325  =   .  4  X  378  =   .  5  x  306  = 

3x184=.  4x76=.  5x       9  = 


3  X 


4  X     .     = 


5  X 


86    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 


171.  Subtract : 

111  ])iactice  be^in  at  units  and  subtract  thus  : 
8214 

8  from  14  leaves  6. 
1 7«3o 
3  from  10  leaves  7. 

7  from  11  leaves  4. 
1  from  7  leaves  6.       The  difference  is  6476. 

172.  Exercise. 

Copy  and  subtract  : 

/.  832         6.  1000       //.  7001       16.  7200       2f.  3020 
574  _389  2098  _99  1982 

2.  722         7.  1070       12.  3010       17.  3G72       22.  7008 
448  179  2111  908  1999 

3.  320         8.  2080       13.  5002       18.  7202       23.  G320 
107  1979  4983  5193  2198 

4.  250         9.  3420        14.  5000       /5.   8201       24.   3507 
148  2837  3092  7192  2198 

6.  700       10.  9000       /5.  8030       :?0.   3380      25.  4320 
188  1807  1929  1099  1738 

173.  Find  the  sum  of  113.25,  110.48,  and  10.70. 

113.25  Write  the  sums,  placing  dollars  under  dollars  and 

10.48  cents  under  cents.     Add  as  in  ordinary  numbers, 

Q' «6  and  place  the  decimal  point  before  the  last  two 

$24.49  figures  to  separate  dollars  from  cents. 


ADDITION  AND  SUBTRACTION.  S7 


174.  Exercise. 

Find  the 

sum  : 

1. 

2. 

3. 

4. 

%  3.25 

$  8.48 

$12.38 

$24.06 

5.63 

7.35 

15.09 

33.45 

7.42 

0.67 

32.26 

8.37 

12.56 

26.89 

31.56 

15.00 

5.  $7.25  4-  $6.48  +  $15.23  +  $28.16. 

6.  $9.32  +  $17  +  $  0.64  +  $36.99. 

7.  $0.56  +  $1.38  +  $10.49  +  $  0.18  +  $14.38. 

176.  From  $27  take  $14.27. 

Write   dollars   under    dollars    and    cents  under 
$27  00 

cents.     Subtract  as  in  ordinary  numbers,  and  place 

the  decimal    point  before   the  last  two  fissures  to 

separate  dollars  from  cents. 

176.  Exercise. 

Find  the  difference  : 

/.  $4.26         2.  $15.42         5.  $146.38         4.  $96.45 
3.83  9.83  27.94  0.87 


5.  $34.10  -  $19.46     7.  $82.56  -  37.49       9.  $10  -  $  6.48 

6.  $  1.48  -  $  0.79     5.  $  5       -    3.69     10.  $50  -  $42.65 

177.  Exercise. 

/.  A  man  bought  a  lot  of  land  for  $350  and  built  on  it 
a  house  that  cost  $1875  ;  how  much  did  both  cost  him  ? 


88    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS- 


2.  James  had  $2.75  and  earned  $3.75  more  ;  liow  much 
had  he  then  ? 

5.  In  a  certain  town  there  are  35 78  males  and  4423 
females  ;  how  many  people  has  the  town  ? 

4.  Take  11.25  from  15.00. 

5.  I  had  a  1000-mile  ticket,  but  have  used  695  miles 
of  it ;  how  many  miles  may  I  yet  ride  on  it  ? 

6.  A  farmer^s  crop  for  a  certain  year  was  268  bushels  of 
wheat,  308  bushels  of  oats,  564  bushels  of  corn,  and  78 
bushels  of  rye  ;  how  much  grain  did  he  raise  that  year  ? 

7.  Frank  and  John  together  have  $4.85.  If  Frank  has 
$2.90,  how  much  has  John  ? 

8.  A  wagon  loaded  with  coal  weighed  5268  pounds.  If 
the  wagon  weighed  1825  pounds,  what  was  the  weight  of 
the  coal  ? 

9.  A  farmer  bought  a  plow  for  $8.75  and  gave  in  pay- 
ment a  $20-bill  ;  how  much  change  should  he  receive  ? 

10.  A  farmer  was  paid  $175.20  for  his  wheat  and  $106.30 
for  his  potatoes  ;  how  much  was  he  paid  for  both  ? 

//.  The  number  of  ladies  at  a  normal  school  duriiiij  a 
certain  year  was  539  and  the  number  of  men  221 ;  how 
many  were  there  of  both  ?  By  how  many  did  the  former 
exceed  the  latter  ? 

12.  My  brother  earned  $1.75  on  Monday,  $1.60  on 
Tuesday,  $1.85  on  Wednesday,  $1.35  on  Thursday,  and 
$1.75  on  Friday  ;  how  much  did  he  earn  in  tlio  5  days  ? 

13.  I  owed  B  $100  and  gave  him  a  cow  worth  $45.50 
and  the  balance  in  cash  ;  how  much  cash  did  I  give 
him  ? 


MULTIPLICATION,  DIVISION,  MENSURATION.     89 


14.  Find  the  sum  of  one  thousand  four  hundred  eight, 
three  thousand  forty-eight,  and  six  hundred  seventy-six. 

15.  Find  the  sum  of  all  numbers  that  are  greater  than 
1298  and  less  than  1304. 

16.  The  following  table  gives  the  number  of  bushels 
of  grain  bought  by  a  dealer  in  one  week  ;  fdl  in  the 
totals  : 


Grain. 

Mon. 

Taes. 

Wed. 

Thurs. 

Fri. 

Sat. 

Total. 

Wheat 

235 
431 
500 

27 

300 

307 

480 

45 

201 

408 

578 

38 

406 

560 

401 

46 

327 

456 

492 

62 

195 

231 

324 

14 

Corn 

Oats 

Rye  

Total 

Multiplication,  Division,  and  Mensuration. 

178.  Oral  Exercise. 

Count  by  6's  : 


/.  From  6  to  60. 

2.  From  60  to  6. 

3.  From  1  to  61. 

4.  From  2  to  62. 


5.  From  3  to  63. 

6.  From  4  to  64. 

7.  From  5  to  65. 

8.  From  65  to  5. 


Count  by  7's  : 

9.  From  7  to  70. 
10.  From  70  to  7. 
//.  From  1  to  71. 
12.  From  2  to  72. 


13.  From  3  to  73. 

14.  From  4  to  74. 

15.  From  5  to  75. 

16.  From  6  to  76. 


90    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 


179.  Exercise. 

Copy,  supplying  the  missing  numbers  : 


5x1  +  1=   . 

5x4+4= 

5x7  +  7=   . 

G  X  1  =  .   - 

6x4  = 

6x7=. 

5x2  +  2=   . 

5x5+5= 

5x8  +  8=  . 

6x2=. 

6  X  5=  « 

6x8=. 

5x3+3=- 

5x6+6= 

5x9  +  9=   . 

6x3=. 

6x6=, 

6x9=. 

180.  Exercise. 

Copy,  supplying 

the  missing  numbers  and  learn  : 

TABLE. 

6x1=. 

6x4=. 

6x7=. 

6x2=. 

6x5=. 

6x8=- 

6x3=. 

6x6=. 

6x9=. 

181.  Exercise. 

Copy,  supplying  the  missing  numbers 


6x1+1=. 

6x4+4= 

6x7+7=. 

7x1=. 

7x4  = 

7x7=. 

6x2  +  2=  . 

6x5+5= 

6x8  +  8=   . 

7x2=. 

7x5=. 

7x8=. 

G  X  3  +  3  =  . 

6x6+6= 

6x9  +  9=   - 

7x3=. 

7x6=. 

7x9=. 

182.  Exercise. 

Copy,  supplying  the  missing  numbers  and  learn  : 
7x1=.  7x4=.  7x7  = 

7x2=.  7x5=.  7x8= 

7x3=.  7x6=..  7x9= 


MULTIPLICATION,  DIVISION,  MENSURATION.     91 


183.  Oral  Exercise. 

Eead,  supplying 

the  missing  numbers  : 

/.  6  X  3  =  . 

19. 

• 

X  6  =  42 

37. 

•     X 

1  =  6 

2.  7  X  7  =  . 

20. 

6 

X  2=  . 

38. 

6  X 

4=  . 

5.  7  X  .   =  49 

21. 

• 

X  4  =  24 

39. 

7  X 

.   =42 

4.  7  X  9  =  . 

22. 

7 

X  .   =21 

40. 

7  X 

.   =63 

5.  7x4=. 

23. 

6 

X  5=  . 

41. 

6  X 

.  =  12 

ff.  6  X  8  =  . 

24. 

• 

X  2  =  14 

42. 

•      X 

7  =  42 

7.  7  x  .   =  28 

25. 

. 

X  6  =  36 

43. 

6  X 

6  =  . 

8.    .  X  7  =  49 

26. 

. 

X  8  =  56 

44. 

6  X 

.  =24 

5.  6  X  7  =  . 

27. 

6 

X  1  =   . 

45. 

7  X 

7=  . 

10.    .  X  2  =  12 

28. 

6 

X  .   =42 

46. 

6  X 

.   =30 

//.  6  X  .   =  36 

29. 

6 

X  9=   . 

47. 

6  X 

.   =6 

/2.  7  X  5  =  . 

30. 

6 

X  .   =  54 

48. 

7  X 

•  =  35 

/5.  7  X  6  =  . 

31. 

. 

X  9  =  54 

49. 

•      X 

3  =  18 

14.    .  X  8  =  48 

32. 

• 

X  9  =  63 

50. 

•     X 

5  =  35 

15.    .  X  4  =  28 

33. 

. 

X  1  =  7 

51. 

7  X 

2  =  . 

/e.  7  X  2  =  . 

34. 

6 

X  .   =  18 

52. 

•     X 

3  =  21 

17.    .  X  5  =  30 

35. 

7 

X  1  =   . 

53. 

6  X 

.  =48 

/5.  7  X  .   =  56 

36. 

7 

X  8=  . 

54. 

7  X 

3=  . 

184.  Exercise.  ' 

1.  Find  the  cost  of  these  groceries  together  : 

6  pounds  of  sugar  at  5  cents  a  pound. 
2  pounds  of  crackers  at  5  cents  a  pound. 
2  quarts  of  vinegar  at  4  cents  a  quart. 

2.  In  a  room  there  are  7  rows  of  desks  and  in  each  row 
8  desks  ;  how  many  desks  are  there  in  all  ? 

5.  A  man  who  sleeps  9  hours  a  day  sleeps  how  long  each 
week? 


92    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 


4.  A  farmer  bought  6  lambs  at  4  dollars  apiece  and  sold 
them  at  5  dollars  apiece  ;  how  much  did  he  make  on  each 
lamb  ?     How  much  on  the  6  lambs  ? 

5.  In  6  bushels  there  are  —  pecks. 

6.  In  7  pecks  there  are  —  quarts. 

7.  Mr.  Brown  gave  3  boxes  of  strawberries  at  8  cents  a 
box  for  syrup  at  6  cents  a  quart  ;  how  many  quarts  did  he 
receive  ? 

8.  A  lady  spends  6  cents  a  day  for  milk ;  how  much  is 
that  a  week  ? 

9.  Show  by  dots  that  6  x  3  —  18. 

10.  \  dozen  pigeons  will  make  how  many  pair  ? 

//.  How  many  bushels  will  7  two-bushel  bags  hold  ? 
12.  How  many  pencils  are  there  in  7  boxes,  if  each  box 
contains  6  pencils  ? 

185.  Each  of  the  six  equal  parts  of  a  number  is 
called  one  sixth  {\)  of  it. 

188.  Each  of  the  seven  equal  parts  of  a  number  is 
called  one  seventh  (|)  of  it. 


187.  Exercise. 

Copy,  writing  the 

proper  figures  in 

place  of  x^: 

/.  ^of    6  =  x 

7.   J  of  42  =  X 

13.  \  of  28  = 

X 

2.   i  of  12  :=  x 

8.   \  of  48  =  X 

U.  ^  of  35  = 

X 

5.  \  of  18  =  X 

9.  i  of  54  =  X 

15.  ^  of  42  = 

X 

4.   J  of  24  =  X 

10.    \oi    7  =  X 

16.   1  of  49  = 

X 

5.  \  of  30  =  X 

//.  |of  14  =  X 

n.  \  of  50  = 

X 

6.  i  of  36  =  X 

12.  |of  21  =x 

18.   \  of  63  = 

X 

MULTIPLICATION,  DIVISION,  MENSURATION.     93 

188.  Oral  Exercise. 

Read  Exercise  187,  supplying  the  missing  numbers. 

189.  Exercise. 

Copy,  supplying  the  missing  numbers  : 

/.  12  =  —  twos.  7.  30  =  —  fives.  13.  63  =  —  nines. 

2.  7  =  —  ones.  8.  24  =  —  fours.  14.  4'^  =  — sevens. 

3.  14  =  —  twos.  9.  48  =  —  eights.  15.  35  =  —  fives. 

4.  18  =  —  threes.  10.  42  —  —  sixes.  16.  36  =  —  sixes. 

5.  6  =  —  ones.  //.  28  =  —  fours.  /7.  4C  =  — sevens. 

6.  21  =  —  threes.  12.  54  =  —  nines.  18.  56  —  — eights. 

190.  Oral  Exercise. 

Read  Exercise  189,  supplying  the  missing  numbers. 

191.  Exercise. 

/.  Divide  12  cards  equally  among  6  boys.  What  part 
of  them  will  each  boy  receive  ?  How  many  will  each  boy 
receive  ? 

2.  How  many  yards  of  carpet  are  there  in  a  piece  21 
feet  long  ? 

3.  48  girls  are  marching  6  in  a  row  ;  how  many  rows 
are  there  ? 

4.  How  many  5-cent  pieces  should  I  get  in  exchange 
for  30  cents  ? 

5.  3  gallons  of  syrup  will  fill  how  many  2-quart  jars  ? 

6.  How  long  will  3  pecks  of  oats  last  a  horse  that  eats 
4  quarts  a  day  ? 


94   WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 

7.  In  28  days  there  are  —  weeks. 

8.  At   4   for   a  cent,   how   much   must  I   pay   for   28 
rivets  ? 

9.  How  many  yards  of  muslin  at  8  cents  a  yard  can  be 
bought  for  8  quarts  of  plums  at  G  cents  a  quart  ? 

10.  A  boy  earned  54  dollars  in  six  months  ;  how  much 
was  he  paid  a  month  ? 

11.  A  dozen  will  make  how  many  pair  ? 

12.  How  many  working  days  are  there  in  a  week  ? 

13.  A  man  who  earns  3  dollars  each  working  day  earns 
how  much  in  a  week  ? 

14.  6  pounds  of  lard  at  8  cents  a  pound,  and  how  many 
cents,  will  pay  for  a  50-cent  bucket  ? 


192.     Oral  Exercise. 

Count  by  8's  : 

/.  From  8  to  80. 

2.  From  80  to  8. 

3.  From  1  to  81. 

4.  From  2  to  82. 

5.  From  3  to  83. 


6.  From  4  to  84. 

7.  From  5  to  85. 

8.  From  6  to  86. 

9.  From  7  to  87. 
10.  From  8  to  88. 


Count  by  9's  : 

//.  From  9  to  90. 

12.  From  90  to  9. 

13.  From  1  to  91. 

14.  From  2  to  92. 

15.  From  3  to  93. 


re.  From  4  to  94. 

17.  From  5  to  95. 

18.  From  6  to  96. 

19.  From  7  to  97. 

20.  From  8  to  98. 


MULTIPLICATION,  DIVISION,  MENSURATION.      95 


193.  Exercise. 

Copy,  supplying 

the  missing  numbers 

7x1  +  1=. 

7x4+4=. 

7x7+7=. 

8  X  1  r=  . 

8x4=. 

8x7=  • 

7x2  +  2=  . 

7x5  +  5=  . 

■7x8  +  8=  . 

8x2=. 

8x5=. 

8x8=. 

7x3  +  3=  . 

7x6  +  6=   . 

7x9  +  9=  . 

8x3=. 

-8x6=. 

8x9=. 

194.  Exercise. 

Copy,  supplying 

the  missing  numbers, 

TABLE. 

and  learn  : 

8x1=. 

8x4=. 

8x7=. 

8x2=. 

8x5=. 

8x8=. 

8x3=. 

8x6=. 

8x9=. 

195.  Exercise. 

Copy,  supplying 

the  missing  numbers 

8x1  +  1=   . 

8x4  +  4=  . 

8x7  +  7=  . 

9x1=. 

9x4=. 

9x7=. 

8x2  +  2=  . 

8x5  +  5=   . 

8x8  +  8=   . 

9x2=. 

9x5=. 

9x8=. 

8x3  +  3=  . 

8x6  +  6=   . 

8x9  +  9=   . 

9x3=. 

9x6=. 

9x9=. 

196.  Exercise. 

Copy,  supplying 

the  missing  numbers, 

TABLE. 

and  learn  : 

9x1=. 

9x4=. 

9x7=. 

9x2=. 

9x5=. 

9x8=. 

9x3=. 

9x6=. 

9x9=. 

96    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 


197.  Oral  Exercise. 

Head,  supplying  the  missing  numbers  : 


/. 

8 

X 

2=  . 

19. 

9 

X 

.=  63 

37. 

. 

X 

1  =  8 

2. 

. 

X 

7  =  G3 

20. 

• 

X 

9  =  81 

38. 

. 

X 

5  =  45 

3. 

• 

X 

1  =  9 

21. 

9 

X 

.=  36 

39. 

9 

X 

.=  72 

4. 

8 

X 

8=  . 

22. 

9 

X 

5  =  . 

40. 

8 

X 

7=  . 

5. 

9 

X 

7=  . 

23. 

8 

X 

.  =  56 

41. 

9 

X 

9=  . 

6. 

. 

X 

9  =  72 

24. 

9 

X 

3=  . 

42. 

• 

X 

6  =  54 

7. 

• 

X 

8  =  72 

25. 

• 

X 

2  =  18 

43. 

. 

X 

4  =  36 

8. 

8 

X 

6=  . 

26. 

9 

X 

.=  9 

44. 

9 

X 

.=  27 

9. 

9 

X 

8=  . 

27. 

9 

X 

1  =  . 

45. 

9 

X 

G  =  . 

10. 

8 

X 

.=  72 

28. 

• 

X 

2  =  16 

46. 

• 

X 

6  =  48 

11. 

8 

X 

.  =  32 

29. 

8 

X 

.=  40 

47. 

9 

X 

.=  18 

12. 

8 

X 

.=  IG 

30. 

8 

X 

.  =  24 

48. 

8 

X 

.=  8 

13. 

8 

X 

5=  . 

31. 

8 

X 

1  =  • 

49. 

9 

X 

.=  54 

14. 

• 

X 

8  =  G4 

32. 

. 

X 

4  =  36 

50. 

• 

X 

7  =  56 

15. 

9 

X 

4=  . 

33. 

8 

X 

4    . 

51. 

9 

X 

.  =  81 

16. 

8 

X 

9=  . 

34. 

8 

X 

3=  . 

52. 

• 

X 

3  =  27 

17. 

. 

X 

5  =  40 

35. 

9 

X 

.=  45 

53. 

• 

X 

3  =  24 

18. 

8 

X 

.=  64 

36. 

9 

X 

2=  . 

54. 

8 

X 

.  =  48 

198.  Exercise. 

/.  In  8  gallons  there  are  —  quarts. 

2.  In  8  weeks  there  are  —  days. 

3.  In  8  pecks  there  are  —  quarts. 

4.  8  X  2  +    .   =  9  X  2. 

5.  9  three-bushel  bags  will  hold  how  much  wheat  ? 

6.  8  two-cent  stamps  will  cost  how  much  ? 

7.  At  4  dollars  a  hundred,  what  will  9  hundred  cabbage 
plants  cost  ? 


MULTIPLICATION,  DIVISION,  MENSURATION.     97 

8.  What  should  be  paid  for  9  dozen  herrings  at  8  cents 
a  dozen  ? 

9.  What  should  be  paid  for  8  hundred  rails  at  6  dollars 
a  hundred  ? 

10.  How  many  feet  long  is  a  fishing  line  8  yards  long  ? 

/  /.  How  many  persons  are  seated  in  a  dining-room  at 
6  tables,  if  there  are  8  persons  at  each  table  ? 

12.  A  drover  sold  8  pair  of  mules  and  one  more ;  how 
many  did  he  sell  ? 

13.  After  a  farmer  sold  8  bags  of  wheat  each  holding  3 
bushels,  he  had  5  bushels  left  ;  how  many  had  he  at  first  ? 

14.  In  1  bushel  there  are  —  pecks,  and  in  1  peck  there 
are  —  quarts  ;  then  in  1  bushel  there  are  —  times  — 
quarts,  or  —  quarts. 

15.  Show  that  there  are  8  pints  in  a  gallon. 

199.  Each  of  the  8  equal  parts  of  a  number  is  one 
eighth  (|)  of  it. 

200.  Each  of  the  9  equal  parts  of  a  number  is  one 
ninth  (i)  of  it. 

201.  Exercise. 

Copy,  writing  the  proper  figures  in  place  of  x  : 


/.  i  of    8  =  X 

7.  ^  of  56  =  X 

13.  i  of  36  =  X 

2.  J  of  IG  =  X 

8.  '\  of  64  =  X 

14.  ^  of  45  =  X 

3.  1  of  24  =  X 

9.   \  of  72  =  X 

15.  ^  of  54  =  X 

4.  4  of  32  =  X 

10.  \Qi    9  =  X 

16.  ^  of  63  =  X 

5.  i  of  40  =  X 

//.  -iof  18  =  X 

n.  ^  of  72  =  X 

6.  -^  of  48  ==  X 
7 

12.  i  of  27  =  X 

18.  i  of  81  =  X 

98    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 


202.  Oral  Exercise. 

Read  Exercise  201,  supplying  the  missing  numbers. 

203.  Exercise. 

Copy,  supplying  the  missing  numbers  : 

/.     8  =  — ones.  7.  27  —  — threes.  13.  72  =  — nines. 

2.  18  =  —  twos.  8.  56  =  —  sevens.  14.  04  =  —  oights. 

3.  24  =  —  threes.  9.  45  =  —  fives.  15.  40  =  —  fives. 

4.  32  =  —fours.  10.  36  =  —fours.  16.  63  =  —sevens. 

5.  9  =  —  ones.  //.  48  =  —  sixes.  17.  72  =  —  eights, 
ff.  16  =  — twos.  12.  54  =  — sixes.  18.  81  =  — nines. 

204.  Oral  Exercise. 

Read  Exercise  203,  supplying  the  missing  numbers. 

205.  Exercise. 

/.  18  cents  will  buy  —  2-cent  stamps. 

2.  How  many  8-foot  logs  can  be  cut  from  a  log  40  feet 
long  ? 

3.  Make  and  solve  a  problem  about  ^  of  16  boys. 

4.  800  posts  cost  56  dollars  ;  how  much  was  that  for 
each  hundred  posts. 

5.  My  brother  and  I  bought  9  dozen  tomato  plants  for 
72  cents  ;  how  much  was  that  a  dozen  ?  How  much 
should  he  pay  if  he  took  6  dozen  ? 

6.  A  log  42  feet  long  will  make  a  10-foot  log  and  how 
many  4-foot  logs  ? 

7.  A  grocer  put  40  pounds  of  sugar  in  8  sacks  of  the 
same  size  ;  how  much  did  he  put  in  each  sack  ? 


MULTIPLICATION,  DIVISION,  MENSURATION.     99 

8.  A  farmer  had  20  pigs.  lie  kept  4  of  them  and  sold 
the  rest ;  how  many  pair  did  he  sell  ? 

9.  How  many  5-dollar  bills  should  I  get  for  45  silver 
dollars  ? 

10.  A  farmer  needs  24  pounds  of  twine  to  bind  his 
wheat ;  how  many  3-pound  balls  must  he  buy  ? 

11.  A  woman  sold  a  dealer  chickens  at  8  cents  a  pound  ; 
he  paid  her  72  cents  for  them  ;  how  many  pounds  did  she 
sell  him  ? 

12.  I  wish  to  plant  64  trees  in  8  rows  ;  how  many  must 
I  put  in  a  row  ? 

2      .r     « 


206.  Find  f  of  9. 


T0f9 


Find  f  of  15. 

i  of  15  is  5,  and  f  of  15  is  2  times  5  or  10. 

207.  Oral  Exercise. 
Find: 

/.  f  of    6.  5.  f  of  12.  P.  ^  f  of  28.  13.  f  of  20. 

2.  f  of    3.  6.  f  of  21.  10.  J  of  32.  14.  f  of  30. 

5.  I  of    9.  7.  J  of  12.*  //.  jof24.  /5.  I  of  45. 

4.  f  of  15.  S.  I  of  16.  12.  f  of  10.  16.  I  of  25. 

208.  Exercise. 

/.  J  of  27  ct.  is  —  ct. ;  and  |  of  27  ct.  is  2  times  —  ct., 
or  —  ct. 

2.  What  is  I  of  6  pounds  ? 


100    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 

3.  What  is  I  of  12  quarts  ? 

4.  J  of  16  bushels  is  —  bushels,  and  }  of  16  bushels  is 
3  times  —  bushels,  or  —  bushels. 

5.  Find  J  of  12  oranges. 

6.  Find  f  of  10  cents. 

7.  If  18  cents  was  the  cost  of  3  pounds  of  sugar,  J  of 
18  cents,  or  —  cents,  was  the  cost  of  1  pound,  and  2  times 
—  cents,  or  —  cents,  was  the  cost  of  2  pounds. 

8.  How  much  should  be  paid  for  2  pencils,  if  3  pencils 
cost  15  cents  ? 

9.  A  baker  charged  me  12  cents  for  3  loaves  of  bread  ; 
what  should  he  charge  me  for  2  loaves  ? 

10.  If  3  horses  are  fed  24  quarts  of  oats  a  day,  how 
much  does  that  allow  for  2  horses  ? 

11.1  owed  a  16-dollar  store  bill  and  paid  J  of  it ;  how 
much  did  I  pay  ? 

12.  Martha  is  20  years  old  and  May  is  J  as  old  ;  how 
old  is  May  ? 

13.  A  boy  paid  10  cents  for  a  tablet,  and  J  as  much  for 
a  ruler  ;  how  much  did  he  pay  for  both  ? 

14.  If  5  loaves  of  bread  cost  30  cents,  1  loaf  costs  J  of 
30  cents,  or  —  cents  ;  and  4  loaves  cost  4  times  —  cents, 
or  —  cents. 

15.  Wliat  should  I  be  paid  for  3  days'  work,  if  I  was 
paid  15  dollars  for  5  days'  work  ? 

16.  A  boy  was  given  25  words  to  spell ;  he  missed  |  of 
them  ;  how  many  did  he  spell  correctly  ? 

17.  There  are  21  dots  in  three  rows,  each  containing 
the  same  number ;  how  many  dots  are  there  in  2  rows  ? 


MULTIPLICATION,  DIVISION,  MENSURATION.    101 

18.  A  grocer  charged  me  9  cents  for  3  lemons,  bat  one 
of  them  was  bad  ;  how  much  should  he  have  charged  me 
for  the  good  lemons  ? 

19.  I  just  paid  $30  for  3  months'  rent  ;  I  now  wish  to 
pay  2  months'  rent  in  advance  ;  how  much  should  I  pay  ? 

20.  If  4  persons  use  8  pounds  of  meat  a  week,  how  much 
should  5  persons  use  ? 

209.  Exercise. 

Copy  and  multiply  : 
/.  27    6.   19   //.  18  16.   13  21.   128  26.   123 

6      7      8      9       8       8 


2.  39  7.  28  12.   26  17.   24  22.   109  27.   109 

_6  _7     _8  _9     7     8 

3.  58  8.   36  13.   74  18.   37  23.   138  28.   108 

_6  ^     _8  _9     7       9 

4.  66  9.   54  14.   93  19.   55  24.   145  29.   117 

_6  _7     _8  __9     6     8 

5.  79  10.   70  15.   50  20.  86  25.  139  30.   122 

6  7       8  9       6        8 


210.  Multiply  $2.38  by  6. 
$2.38         Multiply  as  in  ordinary  numbers,    and    place  the 
Q     decimal    point   before   the   last   two   figures   of   the 
$14  28     pro^^uct  to  separate  dollars  from  cents. 


'io'S   WHOL^' NUMBERS  AND  FRACTIONAL  PARTS. 


211.  Exercise. 

Multiply : 

/.  $4.28  3.  $0.56 

4  9 


5.  $48.07 
8 


7.  $27.08 
8 


2.  $8.72         4.  $138.40 
7  6 


6.  $94.18 
9 


8.  $125.15 
5 


9.  $3.25  by  5.      //.  $21.38  by  8.     /5.  $38.90  by  7. 
10.  $19.20  by  G.     12.  $75.03  by  9.     14.  $1025.03  by  8. 


212.  Exercise. 

Copy  and  fill  in  the  missing  amounts  in  the  following: 

/.  The  cost  of  : 

6  plum  trees  at  $1.25  each  =  $, 

8  peach  trees  at  $0. 80  each  = 

7  pear  trees  at  $1.10  each  = 

9  apple  trees  at  $0.75  each  = 

Total =i 


The  cost  of : 

4 kegs  of  nails  at  $2.75  each  =  $. . . 

6  bushels  of  clover  seed  at 

$5.75  each =    ... 

7  tons  of  bran  at  $18.  (;0  each  =    ... 

8  tons  of  coal  at  $5.75  each  =  . . . 

Total =$... 


MULTIPLICATION,  DIVISION,  MENSURATION.    103 


3.  The  cost  of  : 

9  sheep  at  16.75  each = 

5  horses  at  $125  per  head.  — 
8  cows  at  $40.50  per  head.  = 
8  hogs  at  $8.75  per  head. .  = 

Total = 


213.  Exercise. 

/.  Find  the  cost  of  6  yards  of  cloth  at  15  cents  a  yard. 

2.  A  car  that  makes  24  trips  a  day  makes  how  many 
trips  a  week  ? 

3.  How  many  hours  are  there  in  a  week  ? 

4.  A  gardener  planted  8  rows  of  trees  ;  he  planted  16 
trees  in  each  row  ;  how  many  trees  did  he  plant  in  all  ? 

5.  A  farmer  sold  8  loads  of  wheat  ;  there  were  25  bushels 
in  each  load  ;  how  many  busiiels  were  there  in  all  ? 

6.  A  bushel  of  potatoes  weighs   60   pounds  ;    find  the 
weight  of  8  bushels  of  potatoes. 

7.  A  bushel  of  corn  weighs  56  pounds  ;  find  the  weight 
of  6  bags  of  corn,  each  holding  3  bushels. 

8.  How  many  eggs  will  9  crates  hold,  if  each  crate  holds 
6  dozen  ? 

9.  How  many  persons  can  be  seated  in  8  rows  of  chairs, 
if  there  are  38  chairs  in  each  row  ? 

10.  How  many  persons  can  be  seated  in  6  cars,  if  there 
are  36  seats  in  each  car  and  2  persons  can  sit  on  each  seat? 

//.  A  dealer  sold  to  each  of  8  men  125  barrels  of  flour 
and  has  50  barrels  left ;  how  many  barrels  had  he  at  first  ? 


104    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 

12.  My  lot  cost  $125,  and  my  house  6  times  as  much ; 
how  much  did  they  both  cost  ? 

214.  Oral  Exercise. 

20        30         40        50        60        70        80        90 
Multiply  each  of  the  numbers  above  the  line 

/.  By  2. 
Thus,  3  times  20  are  40  ;  2  times  30  are  GO ;  etc. 


2.  By  3. 

4. 

By  5.           6. 

By 

7. 

8.  By  9. 

3.  By  4. 

5. 

By  6.           7. 

By 

8. 

215.  Oral  Exercise. 

Multiply  : 

/.  25  by  3. 

Thus,  3  times  20 

are 

60  and  3  times  5  are  15 ; 

60  and  15  are  75. 

2.  32  by  2. 

9.  43  by  5. 

16. 

02  by  7. 

3.  48  by  2. 

10.  38  by  5. 

17. 

84  by  8. 

4.  34  by  3. 

//.  QQ  by  5. 

18. 

15  by  8. 

5.  25  by  3. 

12.  72  by  6. 

19. 

30  by  8. 

6.  43  by  4. 

13.  43  by  0. 

20. 

02  by  9. 

7.  50  by  4. 

14.  25  by  6. 

21. 

2b  by  8. 

8.  QQ  by  4. 

15.  75  by  6. 

22. 

30  by  8. 

216.  Exercise. 

/.  AVhat  is  the  cost  of  2  handkerchiefs  at  18  cents  each  ? 

2.  What  is  the  cost  of  3  hatchets  at  30  cents  each  ? 

3.  How  many  boxes  of  rivets  are  there  in  6  packages, 
each  package  containing  a  dozen  boxes  ? 

4.  How  many  yards  of  picture  cord  are  there  in  4  coils, 
each  containing  25  yards  ? 


MULTIPLICATION,  DIVISION,  MENSURATION.    105 

5.  A  lady  bought  2  yards  of  gingham  at  18  cents  a 
yard;  how  much  change  should  she  receive  from  50 
cents  ? 

There  are  16  oimces  {oz.)  in  a  pound  {lb.)  avoirdupois. 

6.  How  many  ounces  are  there  in  5  pounds  avoirdupois  ? 

7.  How  many  ounces  are  there  in  3  pounds  2  ounces  ? 

8.  4  strips  of  carpet,  each  8  yards  long,  are  needed  to 
carpet  a  certain  floor  ;  how  many  yards  are  required  ? 

9.  Find  the  cost  of  3  yards  of  muslin  at  12  cents  a  yard, 
and  3  yards  of  flannel  at  20  cents  a  yard. 

10.  How  much  should  a  trader  pay  for  2  dozen  eggs  at 
15  cents  a  dozen  and  3  pounds  of  butter  at  20  cents  a 
pound  ? 

//.  How  many  are  4  score  and  10  ? 

12.  How  many  are  a  half  dozen  dozen  ? 

13.  How  much  should  a  bushel  of  potatoes  cost  at  15 
cents  a  peck  ? 

14.  How  much  is  received  for  a  gallon  of  cream  at  16 
cents  a  quart  ? 

15.  If  a  peck  of  clover  seed  weighs  15  pounds,  what 
does  a  bushel  weigh  ? 

16.  The  distance  between  two  towns  is  12  miles  ;  how  far 
docs  a  motorman  ride  each  day,  if  he  makes  4  round  trips 
on  a  trolley  line  connecting  these  towns  ? 

217.  3  rows  of  4  dots  each  contain  #^00 
as  many  dots  as  4  columns  of  3  dots  #  •  #  ^ 
each,  which  shows  that  3x4  =  4x3.     #••• 


106    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 

218.  Oral  Exercise. 

Multiply  : 

/.  2  by  20.  Thus,  20  times  2  equals  2  times  20,  or  40. 

2.  2  by  25.  6.  3  by  60.  10.  4  by  60.  14.  6  by  48. 

3.  2  by  40.  7.  3  by  25.  //.   5  by  25.  15.  7  by  14. 

4.  2  by  26.  8.  3  by  42.  12.  5  by  50.  16.  8  by  24. 

5.  2  by  54.  9.  3  by  18.  13.  0  by  36.  17.  9  by  31. 

219.  Exercise. 

/.  A  boy  sold  24  newspapers  at  2  cents  each  ;  how  much 

did  he  receive  for  them  ? 

Since  he  sold  one  newspaper  for  2  cents,  he  sold  24  for  24  times 
2  cents,  which  equals  2  times  24  cents,  or  —  cents. 

2.  Find  the  cost  of  20  two-cent  postage  stamps. 

3.  A  trolley  car  conductor  collected  on  one  trip  5  cents 
from  each  of  18  persons  ;  how  much  did  he  collect  ? 

4.  How  many  persons  are  seated  at  16  tables,  if  there 
are  4  persons  at  each  table  ? 

5.  How  many  days  are  there  in  21  weeks  ? 

6.  How  much  does  it  cost  to  ride  28  miles  on  a  railroad, 
if  the  fare  is  3  cents  a  mile  ? 

7.  If  school  is  in  session  6  hours  a  day  for  20  days  of  a 
month,  how  many  hours  is  it  in  session  in  a  month  ? 

8.  Find  the  gain  on  a  dozen  tablets  bought  for  50  cents 
and  sold  at  6  cents  each. 

220.  Find  the  cost  of  126  yards  of  cloth  at  5^  a  yard. 
$1.26  126  X  5^  =the  cost. 

5  But  126  X  5^  =  5  X  126^,  or  5  x  |1.26. 

rrri  Therefore,  5  x  $1.26,  or  $6.30  =  the  cost. 

In  practice,  multiply  by  the  smaller  mtn^>er. 


MULTIPLICATION,  DIVISION,  MENSURATION.    107 

221.  Exercise. 

Find  the  cost  of  : 

/.  126  bushels  of  seed  at  15  per  bushel. 

2.  228  pounds  of  soap  at  5^  per  pound. 

3.  476  pounds  of  sugar  at  6^  per  pound. 

4.  208  quarts  of  milk  at  6^*  per  quart. 

5.  165  barrels  of  apples  at  $4  per  barrel. 

6.  125  yards  of  carpet  at  $2  per  yard. 

7.  288  fruit  trees  at  $1  each. 

8.  385  gallons  of  oil  at  90  per  gallon. 

9.  A  bushel  of  chestnuts  at  8^  a  quart. 

10.  75  cans  of  corn  at  8^  each. 


222.  Exercise. 

Copy  and  divide  : 

/.  Q)Q^ 

10.  7)84 

19.  6)954 

25.  8)624 

2.  6)60 

//.  8)96 

20.   6)750 

29.  8)920 

3.  7)77 

'     12.  7)98 

21.  7)147 

50.  8)704 

4.  7)70 

13.  6)102 

22.  7)791 

31.  9)819 

5.  8)88 

14.  9)108 

23.   7)812 

52.   9)765 

6.  6)84 

15.  8)104 

24.   7)903 

55.   9)648 

7.   6)90 

16.  6)366 

25.  7)814 

34.   9)531 

8.  6)96 

17.  6)672 

26.  8)248 

55.  9)396 

9.   7)91 

18.  6)804 

27.  8)896 

36.  9)612 

223.  Divide  $18.75  by  5. 
5  "i  118  75  Divide  as  in  ordinary  numbers,   and   place 

"T       IT  the  decimal  before  the  last  two  figures  to  sepa- 

rate dollars  from  cents. 


108    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 

224.  Exercise. 

Divide  : 

/.  2)118.72  5.  y)$35.14  9.  $178.50  by  5 

2.  3)$31.05  6.  9)1193.05  10.  $374.24  by  4 

3.  6)115.48  7.  5)1100.10  //.  $127.00  by  5 

4.  8)133.60  8.  6)$127.08  12.  $38.07  by  9 

225.  Exercise. 

Find  the  cost  of  1  pound  of  each,  if : 
/.  6  lb.  of  coffee  cost  $2.10. 
2.  8  lb.  of  dried  beef  cost  $2.00. 
5.  5  lb.  of  ham  cost  $0.80. 

4.  4  lb.  of  tea  cost  $2.80. 

5.  7  lb.  of  butter  cost  $1.96. 

6.  9  lb.  of  cheese  cost  $1.44. 

7.  3  lb.  of  yarn  cost  $1.05. 

8.  8  lb.  of  prunes  cost  $1.20. 

226.  Exercise. 

/.  A  lady  bought  6  yards  of  cloth  for  96  cents  ;  how 
much  was  it  a  yard  ? 

2.  A  dealer  sold  6  horses  for  $900 ;  he  received  the 
same  sum  for  each  ;  how  much  did  he  receive  for  each  ? 

3.  7  men  caught  819  shad  and  shared  them  equally  ; 
how  many  did  each  receive  ? 

4.  A  man  earned  $84  in  6  weeks  and  his  son  earned  $66  ; 
liow  much  did  both  earn  ?  How  much  did  both  earn  in 
a  week  ? 


MULTIPLICATION,  DIVISION,  MENSURATION.     109 

5.  A  man  rode  644  miles  on  Tuesday  and  \  as  far  on 
Wednesday  ;  how  far  did  he  ride  in  the  two  days  ? 

6.  In  9  class-rooms  there  are  405  desks  ;  if  each  room 
contains  the  same  number  of  desks,  how  many  are  there 
in  each  room  ? 

7.  A  dealer  had  500  dressed  chickens  ;  he  sold  68  of 
them  and  shipped  the  others  in  6  barrels  of  equal  size  ; 
how  many  did  he  put  in  each  barrel  ? 

8.  Divide  818  -  106  by  8. 

9.  I  owed  $718,  and  paid  $32;  I  wish  to  pay  the 
rest  in  6  equal  parts ;  how  much  must  I  pay  in  each 
part  ?  * 

10.  Make  and  solve  a  problem  about  672  trees  that  were 
sold  during  6  days. 

//.  Divide  386  +  428  -  128  by  7. 

12.  Divide  $648  among  3  persons,  giving  the  first 
person  ^  of  the  money,  the  second  ^  of  it,  and  the  third 
the  remainder. 

227.  Exercise. 
Measure : 

/.  96  by  6.  5.  402  by  6.  9.  801  by  9. 

2.  108  by  9.  6.  833  by  7.  10.  726  by  6. 

3.  272  by  8.  7.  504  by  9.  //.  749  by  7. 

4.  896  by  7.  8.  376  by  8.  12.  816  by  8. 

228.  Measure  $12.05  by  $0.05. 

$13.05  :  $0.05  =  12050  :  5^  =  1205  --  5,  or  241. 


110    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 

229.  Exercise. 
Measure  : 

/.  $4.24  by  $0.04.        7.  $12.72  by  6^. 

2.  $5  by  5^'.  8.   $0.96  by  4^. 

3.  $18.36  by  $0.09.       9.   $12.50  by  5^. 

4.  $12.32  by  $0.08.  /O.  $12.25  by  5^. 

5.  $50  by  $0.05.  //.  $81  by  9^. 

6.  $27.99  by  $0.09.  12,  $120  by  80. 

230.  Exercise. 

Find  how  many  can  be  bought : 
/.  2-cent  postage  stamps  for  $1.72. 

2.  5-cent  tablets  for  $11.65. 

3.  3-cent  pencils  for  $0.84. 

4.  5-ceut  loaves  of  bread  for  $2.40. 

5.  4-cent  boxes  of  matches  for  $0.60. 

6.  8-cent  spools  of  thread  for  $1.12. 

7.  8-cent  note  books  for  $3.44. 

8.  5-cent  spools  of  thread  for  $1. 

231.  Exercise. 

/.  llow  many  pecks  are  there  in  216  quarts  ? 

2.  How  many  weeks  are  there  in  364  days  ? 

3.  How  many  trains  of  9  cars  each  can  be  made  up 
from  189  cars  ? 

4.  How  many  weeks  will  it  take  to  get  $344  by  saving 
$8  a  week  ? 

5.  Find  the  cost  of  108  marbles,  if  9  are  bought  for  a 
cent. 


MULTIPLICATION,  DIVISION,  MENSURATION.     1 1 1 

6.  How  many  8-cent  stamps  can  you  buy  for  96  cents  ? 

7.  How  long  will  it  take  a  boat  to  sail  224  miles,  if  it 
sails  8  miles  an  hour  ? 

8.  If  coal  costs  $6  a  ton,  how  many  tons  can  be  bought 
for  $216  ? 

9.  How  many  groups  of  6  marbles  each  can  be  made 
with  150  marbles  ? 

10.  A  dealer  bought  8  dozen  hats  in  boxes  of  6  hats 
each  ;  how  many  boxes  were  there  ? 

//.  A  dealer  bought  750  cigars,  and  after  selling  324  of 
them,  put  the  rest  in  packages  of  6  each  ;  how  many 
packages  had  he  ? 

12.  If  the  distance  around  a  wheel  is  6  feet,  how  many 
times  will  it  turn  in  going  882  feet  ? 

13.  144  trees  can  be  planted  in  rows  of  6  trees  each,  8 
trees  each,  or  9  trees  each  ;  how  many  rows  of  each  kind 
would  they  make  ? 


232.  Exercise. 

/.  What  is  each  of  the  three  equal  parts 
of  1  called  ? 

2.  How  many  thirds  are  there  in  1  ? 
r         3.  Read  :  i  ;  f  ;  |. 

4.  What  is  each  of  the  six  equal  parts  of 
1  called  ? 

5.  What  are  2  of  the  six  equal  parts  of  1 
called  ? 


112    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 

6.  Kead :  J ;  ^  ;  f  ;  S ;  I ;  f 

7.  In  J  there  are  how  many  sixths  ? 

8.  In  f  there  are  how  many  sixtlis  ? 

9.  In  J  there  are  how  many  sixths  ? 

10.  Show  that  ^  of  2  is  |,  or  f 
//.  Show  that  ^  of  3  is  J,  or  J. 
12.  Show  that  ^  of  4  is  ^,  or  f. 
/3.  Show  that  J  of  5  is  |, 

233.  Find  ^  of  26. 

Answer  thus  :  ^  of  26  is  4,  and  3  over  ;  ^  of  2  is  |,  or  §. 
The  answer  is  2^. 

234.  Oral  Exercise. 

Find  : 

/.  J  of  7.  4.  I  of  13.         7.  J  of  28.         10.  \  of  34. 

2.  ^  of  8.  5.  ^  of  16.         S.   J  of  25.          //.  J  of  45. 

5.  I  of  9.  6.  ^  of  20.         9.  J  of  40.          12.  J  of  50. 

235.  How  many  6's  are  there  in  16  ?     16  =  2|  sixes. 


Supply  the  missing  numbers,  using  drawings  to  illus- 
trate : 
/.     8  =  —  sixes.     4.  15  =  —  sixes.     7.  28  =  —  sixes. 
2.     9  =  —  sixes.     5.  17  =  —  sixes.     8.  33  =  —  sixes. 
5.   14  =  —  sixes.     6.  20  =  —  sixes.     9.  16  =  — sixes. 


MULTIPLICATION,  DIVISION,  MENSURATION.     113 


237.  Exercise. 

/.  If  G  pounds  of  soap  cost  25  cents,  how  much  is  the 
soap  a  pound  ? 

2.  A  dealer  has  50  handkerchiefs  which  he  wishes  to 
put  up  in  boxes  of  half  a  dozen  each  ;  how  many  boxes 
will  there  be,  and  how  many  handkerchiefs  over  ? 

3.  What  is  each  of  the  seven  equal  parts  of  1  called  ? 

4.  Show  that  i  of  2  is  f . 

5.  Read  :  f  ;  f ;  4  :  f  ;  f  ;  -I- 

6.  Show  that  there  are  2f  sevens  in  16. 

7.  Find  \  of  each  number  in  the  outer  ring  of  the  left- 
hand  design. 


8.  Find  how  many  7's  there  are  in  each  number  in  the 
outer  ring  of  the  right-hand  design. 

9.  25  days  are  —  weeks  —  days. 

10.  K  man  who  earns  15  dollars  in  the  G  working  days 
of  a  week  earns  how  much  a  day  ? 

//.  January  has  31  days  ;  this  is  how  many  weeks  and 
days  ? 

12.  April  has  30  days ;  this  is  how  many  weeks  and  days? 
8 


114   WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 

13.  When  board  costs  10  dollars  a  week,  how  much  is  it 
per  day  ? 

14.  How  many  pounds  of  sugar  at  6  cents  a  pound  can 
be  bought  for  8  pounds  of  duck  at  7  cents  a  pound  ? 

238.  Oral  Exercise. 


\ 


1.  What  is  each  of  the  8 
equal  parts  of  1  called  ? 

2.  How  many  eighths  are 
there  in  1  ? 

3.  How  many  eighths  are 
there  in  J  ?  in  J  ? 

4.  How  many  eighths  are  there  in  J  ? 

5.  Eead  :  I ;  f ;  I ;  I ;  S ;  |. 

6.  How  many  fourths  are  there  in  f  ?  in  ^  ?  in  |  ? 

7.  How  many  halves  are  there  in  ^  ?  in  f  ? 

8.  What  is  each  of  the  9  equal  parts 
of  1  called  ? 

9.  How  many  ninths  are  there  in  1  ? 

10.  How  many  ninths  are  there  in  \  ? 
inf  ? 

//.  Read:  | ;  4 ;  | ;  | ;  3 ;  4  ;  i ;  i 
12.  How  many  thirds  are  there  in  ^  ?  ^  ? 


1 
..............1 

1 

9 

J ! 

1        f 
j        I 

239.  Exercise. 

/.  Show  that  i  of  2  is  |,  or  J. 

2.  Show  that  i  of  4  is  |,  or  J. 

3.  Show  that  ^  of  6  is  |,  or  J. 

4.  Show  that  -^  of  3  is  ^,  or  J. 

5.  Show  that  i  of  G  is  i  or  |. 


MULTIPLICATION,  DIVISION,  MENSURATION.     115 


240.  Oral  Exercise. 

What  is  : 

/.  iof    5?  5.  ^  of  30?  //.  iof  25  ? 

2.  ^  of  25  ?  7.  ^  of  35  ?  12.  1  of  30  ? 

5.  i  of  20  ?  8.  \  of  42  ?  /5.  ^  of  38  ? 

4.  ^  of  22  ?  5.  ^  of  50  ?  /4.  ^  of  40  ? 

5.  ^  of  18  ?             10.  I  of  12  ?  15.  ^  of  53  ? 

241.  Exercise. 

/.  Show  that  there  are  2 J  eights  in  20. 

2.  In  15  quarts  there  are  —  pecks  —  quarts. 

3.  In  25  quarts  there  are  —  pecks  —  quarts. 

4.  Show  that  there  are  2f  nines  in  20. 

5.  Show  that  there  are  2^  nines  in  21. 

6.  If  a  peck  of  plums  sold  for  50  cents,  how  much  were 
they  a  quart  ? 

7.  25  cents  will  pay  for  how  many  pounds  of  rice  at  8 
cents  a  pound,  and  how  many  cents  will  be  left  over  ? 

8.  38  dots  will  make  how  many  lines  of  9  dots  each, 
and  how  many  dots  will  be  over? 

242.  Exercise. 

/.  How  many  days  are  there  in  G  weeks  ? 
2.  30  quarts  are  —  pecks  —  quarts. 
5.  2  sets  of  casters,  4  in  a  set,  cost  20  cents  ;  how  much 
did  each  caster  cost  ? 

4.  Find  the  gain  on  40  papers  bought  at  1  ct.  each  and 
sold  at  2  ct.  each. 

5.  Since  16  ounces  make  a  pound,  how  many  pounds 
will  2  cakes  of  soap  weigh,  if  each  cake  weighs  8  ounces  ? 


116    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 


6.  How  old  is  a  man  who  was  40  years  old  16  years  ago  ? 

7.  A  trolley  car  that  makes  2  round  trips  an  hour  over 
a  road  3  miles  long,  runs  how  far  each  hour  ? 

8.  Supply  the  missing  numbers  : 

1  —  t  —  t  =  t  =  t  =  ff  =  f  —  ¥  =  t" 

9.  Since  —  pints  make  a  quart,  and  —  quarts  make  a 
gallon,  there  are  —  times  —  pints,  or —  pints,  in  a  gallon. 

10.  At  8  cents  a  pint,  what  is  the  selling  price  of  a  gal- 
lon of  cream  ? 

11.  A.  dime,  a  5-cent  piece,  and  a  one-cent  piece  are 
together  worth  how  much  less  than  2  dimes  ? 

12.  How  many  crates  does  a  market  man  require  to  pack 
54  dozen  eggs,  if  he  puts  6  dozen  in  a  crate  ? 

13.  —  pints  make  a  gallon;  how  many  gallons  are 
there  in  64  pints  ? 

243.  Oral  Exercise. 
Find  : 

/.  f  of  12.  6.  f  of  35.  //.  4  of  70.  16.  |  of  48. 

2.  I  of  18.  7.  f  of  28.  12.  |  of  16.  17.  I  of  80. 

3.  I  of  30.  8.  4  of  21.  13.  I  of  32.  18.  f  of  27. 

4.  I  of  24.  9.  4  of  42.  14.  |  of  40.  19.  ^  of  36. 

5.  f  of  14.  10.  f  of  49.  15.  I  of  24.  20.  \  of  18. 

244.  Exercise. 

/.  If  0  times  a  number  is  30,  the  number  is  what  part 
of  30  ?    What  is  the  number  ? 

2.  If  0  times  a  number  is  24,  the  number  is  —  of  24,  or 
—  ;  and  5  times  the  number  is  —  times  — ,  or  — . 


MULTIPLICATION,  DIVISION,  MENSURATION.    117 

3.  If  7  times  a  number  is  56,  what  is  6  times  the  number? 

4.  ^^  of  a  score  are  how  many  ? 

5.  A  farmer  had  63  sheep  and  sold  ^  of  them  ;  how  many 
had  he  left  ? 

6.  I  of  a  week  is  f  of  —  days,  or  —  days. 

7.  A  turkey  weighing  13  pounds,  live  weight,  weighed 
only  I  as  much  when  dressed  ;  how  much  did  it  lose  in  the 
dressing  ? 

8.  7  cans  of  lard  of  equal  size  weighed  70  pounds  ;  how 
much  did  2  of  them  weigh  ? 

9.  If  6  dozen  bananas  cost  60  cents,  what  should  3  dozen 
cost? 

10.  If  6  hundred  rails  sold  for  48  dollars,  2  hundred 
should  sell  for  —  dollars. 

//.  I  have  42  tablets  in  7  packages  of  equal  size  ;  how 
many  tablets  are  there  in  3  of  these  packages  ? 

12.  A  miner  earned  $30  a  month  and  his  son  ^V  ^^ 
much  ;  how  much  did  both  earn  a  month  ? 

13.  If  a  turkey  weighing  9  pounds  sold  for  90  cents, 
what  should  one  weighing  8  pounds  sell  for  ? 

14.  In  taking  8  steps  a  man  walked  24  feet ;  how  far 
had  he  walked  when  he  had  taken  6  steps  ? 

15.  A  father  gave  each  of  his  3  sons  ^  of  a  farm  of  56 
acres  ;  how  many  acres  had  he  left  ? 

16.  If  8  times  a  number  is  32,  what  is  7  times  the 
number  ? 

17.  A  of  90  cents  is  —  cents,  or  —  dimes. 

18.  A  carpenter  bought  18  hinges  for  9  doors  ;  how  many 
will  he  use  to  hang  4  doors  ? 


1 1 8    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 

246.  Exercise — Miscellaneous  Problems. 

/.  Write  the  letters  used  in  the  Roman  system  of  nota- 
tion and  the  number  for  which  each  stands. 

2.  Express  by  the  Roman  system  : 

19;  44;  406;  1909. 

3.  Express  in  figures  : 

XCIV;        MCD;        MCMVI;        CMI ;        VIV. 

4.  In  $7486,  the  7  expresses  how  many  dollars  ?  the 
4  ?  the  8  ? 

5.  87  +  168  +  326  +  308  =  ? 

6.  327  +  109  +  426  +     ?  =  1000. 

7.  1487  -  896  =  ? 

8.  2939  -     ?    =  1000  ? 

9.  ?     -  526  =  879  ? 

10.  K  dealer  bought  seventeen  hundred  forty  bushels 
of  wheat  and  sold  eleven  hundred  ninety  bushels  of  it ; 
how  much  had  he  left  ? 

//•  What  number  is  101  greater  than  999  ? 

12.  A.  man  bought  a  house  costing  $7250  and  paid 
$4750  on  it  ;  how  much  does  he  yet  owe  ? 

13.  A  man  bought  a  vest  for  $2.75  and  a  pair  of  shoes 
for  $3.50  ;  how  much  change  should  he  receive  from  a 
$10-bill  ? 

14.  From  twenty,  dollars  and  seventy-five  cents  take  the 
sum  of  four  dollars  and  twenty  cents  and  six  dollars  and 
eighty-five  cents. 

15.  A  farmer  bought  a  horse  for  $160  and  sold  him  so 


MISCELLANEOUS  PROBLEMS.  119 

as  to  gain  ^  of  this   sum  ;  find  the  selling  price  of  the 
horse. 

16.  There  were  1685  soldiers  in  a  camp,  but  f  of  them 
were  sent  away  ;  how  many  remained  ? 

17.  Which  is  nearer  1000   and   by  how  much,   899  or 
1101? 

18.  From  4275  take  36  hundred. 

19.  How  old  was  my  father  25  years  ago,  if  he  is  74 
years  old  now  ? 

20.  If  the  first  problem  on  a  page  is  the  128th  and  the 
last  the  153d,  how  many  problems  are  there  on  the  page  ? 

21.  Add  down  ;  add  across  : 

448  +  327  +  1320  = 

79  +  108+  768  = 

175  +  927  +  7856  = 

8  +  128  +   96  = 


.    +    .     +       .    = 
22.  Find  the   missing  amounts   by  adding   down  and 
subtracting  across.     See  that  the  answer  to  (7)  is  the  sum 
of  the  amounts  found  for  column  (c). 


(a) 

(b) 

(c) 

(1) 

$6750.30  - 

$1007.50  = 

p 

(2) 

35.35  - 

20.95  = 

? 

(3) 

180.00  - 

178.25  = 

? 

(4) 

3738.00  - 

987.95  = 

p 

(5) 

3753.60  - 

8.85  = 

p 

(6) 

37.85  - 

5 

13.70  = 

? 

(7) 

? 

? 

120    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 

246.  The  following  table,  called  the  Multiplica- 
tion Table,  shows  the  results  obtained  by  taking 
each  of  the  numbers  from  1  to  12,  inclusive,  from  1 
to  12  times. 

multiplication  table. 


1    X 

1=    1 

2x1=2 

3x1=3 

4x1=4 

1    X 

2=    2 

2x2=4 

3x2=6 

4x2=8 

1    X 

3=3 

2x3=6 

3x3=9 

4  X    3  =  12 

1    X 

4=    4 

2x4=8 

3  X    4  =  12 

4  X    4  =  16 

1    X 

5=    5 

2  X    5  =  10 

3  X    5  =  15 

4  X    5  =  20 

1    X 

6=    6 

2  X    6  =  12 

3  X    6  =  18 

4  X    6  =  24 

1    X 

7=    7 

2  X    7  =  14 

3  X    7  =  21 

4  X    7  =  28 

1    X 

8=    8 

2  X    8  =  16 

3  X    8  =  24 

4  X    8  =  32 

1    X 

9=    9 

2  X    9  =  18 

3  X    9  =  27 

4  X     9  =  36 

1    X 

10  =  10 

2  X  10  =  20 

3  X  10  =  30 

4  X  10  =  40 

1    X 

11  =  11 

2  X  11  =  22 

3  X  11  =  33 

4  X  11  =  44 

1    X 

12  =  12 

2  X  12  =  24 

3  X  12  =  36 

4  X  12  =  48 

5  X 

1  =    5 

0x1=6 

7x1=7 

8x1=8 

5  X 

2  =  10 

6  X    2  =  12 

7  X     2  =  14 

8  X    2  =  16 

5  X 

3  =  15 

6  X    3  =  18 

7  X    3  =  21 

8  X    3  =  24 

5  X 

4  =  20 

6  X    4  =  24 

7  X    4  =  28 

8  X    4  =  32 

5  X 

5  =  25 

6  X    5  =  30 

7  X    5  =  35 

8  X    5  =  40 

5  X 

6=30 

6  X    6  =  36 

7  X    6  =  42 

8  X    6  =  48 

5  X 

7  =  35 

6  X    7  =  42 

7  X    7  =  49 

8x    7  =  56 

5  X 

8  =  40 

6  X    8  =  48 

7  X    8  =  56 

8  X    8  =  64 

5  X 

9  =  45 

6  X    9  =  54 

7  X    9  =  63 

8  X    9  =  72 

5  X 

10  =  50 

6  X  10  =  60 

7  X  10  =  70 

8  X  10  =  80 

5  X 

11  =  55 

6  X  11  =  66 

7  X  11  =  77 

8  X  11  =  88 

5  X 

12  =  60 

6  X  12  =  72 

7  X  12  =  84 

8  X  12  =  96 

9  y 

1=     9 

10  X    1  =    10 

11  X     1  =    11 

12  X    1  =    12 

9  X 

2=    18 

10  X    2  =    20 

11  X    2  =    22 

12  X    2=    24 

9  X 

3=    27 

10  X    3  =    30 

11  X    3=    83 

12  X    3  =    36 

9  X 

4=    36 

10  X    4  =    40 

11  X    4=    44 

12  X    4  =    48 

9  X 

5=   45 

10  X     5=    50 

11  X    5=    55 

12  X    5  =    60 

9  X 

6=    54 

10  X    6  =    60 

11  X     6=    66 

12  X    6=    72 

9  X 

7=   63 

10  X    7  =    70 

1 1  X    7  =    77 

12  X     7  =    84 

9  X 

8=    72 

10  X    8  =    to 

11  X    8=    88 

12  X    8  =    96 

9  X 

9=   81 

10  X    9  -    90 

1 1  X    9  ^    99 

12  X    9  =  108 

9  X 

10=    90 

10  X  10=  100 

11  X  10=  110 

12  X  10  =  120 

9  X 

U  =    99 

10  X  11  ■---  110 

11  X  11  =  121 

12  X  11  =  132 

9  X 

12  =  108 

10  X  13  =  120 

11  X  12  =  132 

12  X  12  =  144 

MULTIPLICATION,  DIVISION,  MENSURATION.    121 


247.  10  X  5  =  50.     From  this  we  see  that 

Annexing  a  cipher  to  a  number  multiplies  it 
by  10. 

248.  Oral  Exercise. 

Read,  supplying  the  missing  numbers  : 


/.  10  X  7  = 
2.  10  X  9  = 
5.  10  X  13  = 

4.  10  X  17  = 

5.  10  X  27  = 

6.  10  X  35  = 


7.  10  X  42  = 

8.  10  X  54  = 

9.  10  X  61  = 
10.  10  X  75  = 
//.  10  X  29  = 
12.   10  X  80  =: 


13.  10  X  225  = 

14.  10  X  201  = 

15.  10  X  278  = 

16.  10  X  370  = 

17.  10  X  300  = 

18.  10  X  990  = 


249.  100  X  5  =  5  X  100  =  500.    From  this  we  see 
I      that 

Annexing  two  cipJiers  to  a  number  multiplies  it 
by  100. 

250.  Oral  Exercise. 

Read,  supplying  the  missing  numbers  : 


/.  100  X     7  = 

2.  100  X     9  = 

3.  100  X     2  = 

4.  100  X     6  = 

5.  100  X  21  = 


6.  100  X  15  = 

7.  100  X  35  = 

8.  100  X  20  = 

9.  100  X  48  = 
10.  100  X  99  = 


//.  100  X  70  = 

12.  100  X  10  = 

13.  100  X  25  = 

14.  100  X  72  = 

15.  100  X  69  = 


251.  1000  X  ^  =  ry  X  1000  =  5000.  From  this  we 
see  that 

A  nnexing  three  ciphers  to  a  number  multiplies  it 
by  1000. 


122    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 

262.  Oral  Exercise. 

Read,  supplying  the  missing  numbers  : 
/.   1000  X  2  =  .      4.   1000  X  1  =  .      7.   1000  x  8  =  • 
2.   1000  X  4  ==   .      5.    1000  x  0  =   .      8.   1000  x  3  =   . 
5.  1000  X  7  =   •      6.  1000  X  5  =  .     9.  1000  x  6  =  • 

253.  Exercise. 

/.  100  dimes  are  worth  —  cents. 

2.  At  15  cents  a  quart,  how  much  should  be  received 
for  10  quarts  of  chestnuts  ? 

3.  Find   the   weight   of   10   shad,    if   they  average  8 
pounds  a  pair. 

4.  What  must  be  paid  for  100  postal  cards  ? 

5.  A  drover  bought  100  head  of  steers  at  an  average  of 
$35  a  head  ;  how  much  did  he  pay  for  them  ? 

6.  Find  the  weight  of  1000  two-pound  sacks  of  salt. 

7.  Find  the  cost  of  a  100-acre  farm  at  $20  an  acre. 

8.  How  many  ounces  are  there  in  100  pounds  avoirdu- 


pois ? 


There  are  12  months  (mo.)  in  a  year  (yr.). 


9.  100  years  is  called  a  century  ;  how  many  months 
are  there  in  a  century  ? 

10.  K  keg  of  nails  weighs  100  pounds  ;  find  the  weight 
of  15  kegs  of  nails. 

SUG.     15  X  100  lb.  =  100  X  151b. 

//.  There  are  100  cents  in  a  dollar ;  liow  many  cents  are 
there  in  5  dollars  ? 


MULTIPLICATION,  DIVISION,  MENSURATION.    123 

254.  Multiply  27  by  300. 

27 

Since  300  =  100  x  3, 

300  X  27  =  100  X  3  X  27  =  100  x  81  =  8100. 

8100 

255.  Exercise. 

Find  the  product  of  : 

/.  20  X  15.             5.  50  X  33.  9.  200  x  35. 

2.  30  X  45.             6.  60  X  75.  fO.  200  x  47. 

3.  40  X  62.             7.  70  X  38.  //.  200  x  35. 

4.  40  X  60.             8.  70  X  80.  12.  300  x  28. 

266.  Exercise. 

/.  Find  the  cost  of  40  sewing  machines  at  $28  each. 
2.  Find  the  tuition  paid  by  45  students,  if  each  pays 
1200. 

SuG.     45  X  $200  =  200  x  $45. 

5.  How  many  feet  of  boards  does  a  man  buy,  if  he  buys 
500  boards  each  18  feet  long  ? 

4.  How  much  fertilizer  must  be  bought  for  a  12-acre 
field,  if  200  pounds  are  required  for  an  acre  ? 

6.  A  barrel  of  beef  weighs  200  pounds  ;  find  the  weight 
of  a  shipment  of  17  barrels  of  beef. 

6.  How  many  yards  are  there  in  12  pieces  of  bunting, 
each  containing  50  yards  ? 

7.  There   are   24   sheets  in  a  quire,  and  20  quires  in 
a  ream  ;  how  many  sheets  are  there  in  a  ream  of  paper  ? 

8.  How  much  house  rent  does  a  man  pay  in  3  years, 
if  he  pays  $20  a  month  ? 


124   WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 
267.  Multiply  374  by  24. 


374 

24 


We  first  multiply  374  by  4,  obtaining  1496  for  the  prod- 
uct. We  next  multiply  374  by  2,  obtaining  748  for  the 
1496  product  ;  and  since  the  2  is  3  tens,  we  write  748  so  that 
748  the  8  stands  in  tens'  place.  The  sum  of  these  products, 
QQ7fi      o^  the  entire  product,  is  8976. 

268.  Exercise. 
Copy  and  multiply : 

/.  43  8.  72  15.  59  22.  342  29.  198 

13  87  57  13  39 


64 

9.   79 

f6.   86 

23.   452 

30.   276 

26 

77 

76 

14 

42 

3.   94 

10.   81 

17.   38 

24.   209 

31.   378 

28 

98 

57 

16 

28 

4.   85 

11.   45 

18.   92 

25.   390 

32.   309 

52 

56 

21 

21 

27 

95 

12.   99 

19.   47 

26.   409 

33.   290 

36 

91 

19 

21 

39 

6.   96 

13.   65 

20.   51 

27.  580 

34.   409 

49 

37 

99 

22 

26 

7.   57 

14.   48 

21.   427 

28.   214 

55.  275 

66 

23 

12 

24 

27 

MULTIPLICATION,  DIVISION,  MENSURATION.    125 

259.  Exercise. 

/.  Find  the  weight  of  25  bushels  of  oats,  if  they  weigh 
32  pounds  to  the  bushel. 

2.  How  many  hours  are  there  in  May,  which  has  31 
days  ? 

3.  How  far  will  a  steamboat  run  in  a  day,  if  it  runs  16 
miles  an  hour  ? 

4.  A  barrel  of  flour  weighs  196  pounds  ;  find  the  weight 
of  0  barrels  of  flour. 

5.  Find  the  cost  of  25  thousand  feet  of  boards  at  $32 
per  thousand  feet. 

6.  Find  the  cost  of  16  hundred  fence  posts  at  $18  per 
hundred. 

7.  A  man  left  $8975  to  his  wife  and  4  children,  the 
children  receiving  $1250  each  ;  how  much  did  the  wife 
receive  ? 

8.  A  dealer  bought  28  grain  drills  at  $62  each  and  sold 
them  at  1100  each  ;  find  his  gain. 

SuG.  $100  -  $62,  or  $38  =  the  gain  on  each. 

9.  A  farmer  bought  15  steers  at  $35  a  head,  and  after 
feeding  them  for  6  months  at  a  cost  of  $40  a  montli, 
sold  them  at  an  average  of  $58  a  head ;  find  his 
gain. 

10.  If  a  clerk's  wages  are  $30  a  month,  find  his  wages 
for  a  year. 

/  /.  52  weeks  and  1  day  make  a  common  year ;  how  many 
days  make  a  common  year  ? 

12.  A.  farmer  put  25  loads  of  lime  on  his  lan^,  each 


126    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 

load  averaging  75  bushels  ;  how  much  lime  did  he  put  on 
his  land  ? 

13.  How  many  dozen  eggs  will  24  crates  hold,  if  each 
crate  holds  72  dozen  ? 

260.  Divide  435  by  4. 

i  of  435  =  i  of  (4  hundred  +  3  tens  +  5  units). 
4  )  435  i  of  4  hundred  =  1  hundred. 

108f  i  of  3  tens  =  0  tens,  and  3  tens  over. 

3  tens  +  5  units  =  35  units. 
I  of  35  units  =  8  units,  and  3  units  over, 
i  of  3  units  =  f  of  a  unit.      The  quotient  is  108f, 


261.  Exercise. 

Copy  and  divide  : 

/.  2)37 

10.  5)72 

19.  7)93 

25.  8)500 

2.  2)51 

//.  5)83 

20.  7)99 

29.  8)362 

3.  3)40 

12.  5)99 

21.  7)130 

50.  8)815 

4.  3)47 

13.  5)93 

22.  7)152 

5/.  8)854 

5.  4)57 

14.  6)79 

23.  7)230 

32.  8)963 

6.  4)74 

15.  6)87 

24.   7)136 

55.  9)200 

7.  4)59 

16.  6)92 

25.  8)257 

34.  9)309 

8.  4)94 

17.  6)89 

26.  8)45^ 

55.  9)401 

9.  5)67 

18.  6)94 

27.  8)005 

5tf.  9)118 

MULTIPLICATION,  DIVISION,  MENSURATION.    127 

262.  Exercise. 

/.  When  handkerchiefs  are  sold  at  2  for  25  cents,  how 
much  are  they  apiece  ? 

2.  If  a  turkey  weighing  6  pounds  sold  for  75  cents,  how 
much  was  that  a  pound  ? 

3.  In  2  rods  there  are  33  feet ;  how  many  feet  are  there 
in  1  rod  ? 

4.  How  much  must  be  paid  for  a  pint  of  varnish  at  65 
cents  a  quart  ? 

5.  If  2  dressed  hogs  weigh  625  pounds,  what  is  their 
average  weight  ? 

6.  What  was  the  price  paid  per  acre,  if  a  lot  of  7  acres 
cost  $825  ? 

7.  What  was  the  average  weight  of  2  steers,  if  one 
weighed  875  pounds  and  the  other  964  pounds  ? 

8.  What  is  the  average  age  of  the  five  members  of  a 
graduating  class,  if  their  ages  are  17  years,  16  years,  18 
years,  15  years,  and  15  years,  respectively  ? 

9.  A  farmer  paid  a  boy  $110  for  9  months'  work  ;  what 
were  the  monthly  wages  paid  ? 

10.  A  lady  paid  50  cents  for  4  yards  of  velvet  trimming  ; 
how  much  was  it  a  yard  ? 

//.  Two  cans  of  lard  weighed  50  pounds  apiece  and  a 
third  can  weighed  48  pounds  ;  what  was  their  average 
weight  ? 

263.  Divide  48  by  16. 

^48  ^^^  ^^  ^^  "^  ^'  ^"^^^  16  X  3,  or  3  X  16  r=  48. 


128    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 

264.  Exercise. 

Divide  : 

/.  22  by  11.  18.     G5  by  13.  35.  051  by  93. 

2.  3G  by  12.  19.     54  by  18.  36.  406  by  58. 

5.  28  by  14.  20.     95  by  19.  37.  384  by  48. 

4.  45  by  15.  21.  100  by  50.  38.  423  by  47. 

5.  GO  by  15.  22.  12G  by  G8.  39.  308  by  44. 

6.  GG  by  22.  23.  108  by  3G.  40.  558  by  62. 

7.  60  by  20.  24.   129  by  43.  4/.  305  by  61. 

8.  72  by  24.  25.  124  by  31.  42.  134  by  67. 

9.  81  by  27.  26.  288  by  72.  43.  102  by  34. 
/O.  64  by  16.  27.  324  by  81.  44.  126  by  42. 
//.  72  by  18.  28.  260  by  52:  45.  171  by  57. 

12.  84  by  21.  29.  470  by  94.  46.  192  by  64. 

13.  92  by  23.  50.  174  by  87.  47.  207  by  69. 

14.  78  by  13.  31.  184  by  92.  48.  176  by  88. 
/5.  90  by  18.  32.  228  by  76.  49.  300  by  75. 

16.  96  by  16.  33.  318  by  53.  50.  440  by  55. 

17.  91  by  13.  34.  576  by  96.  51.  140  by  28. 


266.  Divide  795  by  15. 

iV  of  79  is  5,  the  first  figure  of  the  quotient, 
15  )  795  (  53         Taking  15  x  5,  or  75,   from  79,  wo   have  4  re- 
-45  maining. 

45  Annexing  5,  the  next  figure  of  the  dividend,  to 

the  remainder,  we  have  45. 
^^0  of  45  ia  3,the  second  figure  of  the  quotient. 
Taking  15  x  3,  or  45,  from  45,  we  have  notliing  remaining. 
The  quotient  is  53. 


MULTIPLICATION,  DIVISION,  MENSURATION.    129 


266.  Exercise. 

Divide  : 

/.  143  by  11.  10.  656  by  41.  19.  3770  by  65. 

2.  276  by  12.  //.  867  by  51.  20.  4420  by  85. 

3.  429  by  13.  12.  528  by  22.  21.  1804  by  82. 

4.  602  by  14.  13.  1664  by  26.  22.  1932  by  21. 

5.  795  by  15.  U.  2088  by  72.  23.  1098  by  18. 

6.  294  by  21.  15.  1092  by  42.  24.  1736  by  28. 

7.  465  by  31.  16.  1056  by  24.  25.  3752  by  56. 

8.  782  by  23.  17.  2024  by  44.  26.  2538  by  54. 

9.  800  by  32.  18.  2520  by  56.  27.  1961  by  53. 

Measure  : 

28.  252  by  12.     30.     684  by  38.     32.   4000  by  32. 

29.  286  by  22.     31.   1215  by  45.     33.   3591  by  24. 

267.  Exercise. 

/.  A  man  was  paid  a  debt  of  $125  in  25  bills  of  equal 
value  ;  what  was  the  value  of  each  bill  ? 

2.  The  population  of  Railway,  N.  J.,  was  7105  in  1890 
and  7935  in  1900  ;  what  was  the  average  gain  per  year  for 
the  ten  years  ? 

3.  A  teacher  was  employed  at  $810  for  9  months,  but 
he  resigned  at  the  end  of  7  months  ;  how  much  had  he 
earned  ? 

4.  If  a  farm  of  84  acres  is  valued  at  $2100,  what  value 
is  placed  on  each  acre  ? 

5.  What  is  the  average  number  of  miles  that  a  train 

9 


130    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 

must  run  per  hour  to  reach  a  city  990  miles  distant  in  18 
hours  ? 

6.  A  man's  lot  of  25  acres  and  2  cows  are  valued  at  $685  ; 
if  the  cows  are  valued  at  $30  apiece,  what  value  is  placed 
on  the  land  per  acre  ? 

7.  A  man  bought  property  valued  at  $1872;  how  much 
must  he  pay  a  year,  if  he  is  given  12  years  to  pay  for  it 
and  is  to  pay  the  same  amount  each  year  ? 

8.  If  12  acres  of  wheat  yielded  336  bushels,  how  much 
should  25  acres  yield  at  the  same  average  ? 

9.  I  bought  14  thousand  feet  of  white  pine  boards  for 
$448  ;  what  did  I  pay  per  thousand  feet  ? 

10.  A  drover  bought  25  sheep  for  $100  and  sold  them 
for  $150 ;  what  was  the  average  gain  per  head  ? 

263.  Exercise. 

/.  How  many  trips  did  a  farmer  make  to  haul  away  900 
bushels  of  wheat,  if  he  hauled  75  bushels  at  a  load  ? 
900  bushels  was  the  amount  hauled. 
.        ^  75  bushels  was  the  amount  hauled  at  one  load. 

^      900  bu. :  75  bu.  =  900  -f-  75,  or    •  =  the  number  of  trips. 

2.  A  clerk  pays  $25  a  month  rent  for  a  house  valued  at 
$3750  ;  in  how  many  months  will  the  rent  amount  to  the 
value  of  the  house  ? 

3.  A  dealer  shipped  702  dozen  eggs  in  crates  holding  72 
dozen  each  ;  how  many  crates  did  he  ship  ? 

4.  llow  many  feet  are  there  in  1728  inches  ? 

5.  There  are  32  quarts  in  a  bushel ;  how  many  bushels 
are  there  in  608  quarts  ? 


MULTIPLICATION,  DIVISION,  MENSURATION.     131 

6.  How  many  cows  at  125  a  head  can  be  bought  for 
$320,  and  what  amount  will  be  over  ? 

25  )  320  (  12       1320  is  the  amount  to  be  spent. 

25 

$25  is  the  price  paid  for  each  cow. 

f-Q  1320  :  $25  =  320  -f-  25,  or  12,  is  the  number  that 

"2()  can  be  bought,  with  $20  over. 

7.  Corn  weighs  56  pounds  to  the  bushel.  A  farmer's 
crop  when  sold  weighed  7125  pounds ;  how  many  bushels 
had  he,  and  how  many  pounds  over  ? 

8.  At  15  cents  a  yard,  how  many  yards  of  flannel  can  be 
bought  in  exchange  for  9  pounds  of  chickens  at  10  cents 
a  pound  ? 

9.  If  the  distance  around  a  wheel  is  12  feet,  how  many 
times  will  the  wheel  turn  in  going  5280  feet,  or  1  mile? 

269.  Since  10  x  4  =  40,  40  -^  10  =  4,  or  4|0. 
Since  100  x  4  =  400,  400  ^  100  =  4,  or  4|00. 
Since  1000  x  4  =  4000,  4000  -^  1000  =  4,  or  4|000. 
Therefore, 

Cutting  one  cipher  from  the  right  of  a  nwnriber 
divides  the  number  by  10,  cutting  two  ciphers  di- 
vides it  by  100,  three  ciphers  by  1000^  and  so  on. 

270.  Oral  Exercise. 

Read,  supplying  the  missing  numbers: 
/.  30-^10=«  6.    400 -^  100=  .//.  7000 -^  1000  = 
2.     60  -^•  10  =  .  7.  200  -^  100  =  .  12.   8000  -^  1000  = 
5.  90  -^  10  =  o  5.  500  -^  100  =  .  13.   9000  ^  lOOO  = 
4.   500  -^  10  =  .  9.  4000  -^  100  =  •  14.   5000  -^  1000  = 


5.   320  -f-  10  =  .  10.  3200 


100  =:  .  /5.  8000' -^  1000 


132    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 

271.  1200  ^  20  =  60. 

Also,  120|0  -r  6|0,  or  120  -^  2  =  60. 
Therefore, 

Qutting  off  tl\,e  same  number  of  ciphers  from  the 
right  of  two  numbers  does  not  change  tfteir  quotient. 

272.  Oral  Exercise. 

Read,  supplying  the  missing  numbers  : 
/.     40-^20=:.      5.150^30  =  -      5.  3600 -^  400    =• 
2.     80  ^  20  =  •      6.  300  -r-  50  =  •     10.  7200  -^  600    =  • 
5.     90-^30=.      7.350^70=.     //.  8000 -^  4000  =  • 
4.  100  -T-  20  =  .      5.  630  -^  90  =  .     12.  9000  ^  3000  =  • 


Forming,  Reading,  and  Writing   Numbers   Above 
lOOOO;  Fundamental  Operations. 

To  the  teacher.  The  study  of  numbers  above  10000  may  be  de- 
ferred, if  desirable,  and  taken  up  at  the  close  of  Chap.  II  or 
Chap.  IIL 

273.  Thousands  are  counted  up  to  999  thousand. 

274.  A  thousand  thousand  is  called  a  million. 
Millions  are  counted  up  to  999  million. 

275.  A  thousand  million  is  called  a  billion. 
Billions  are  counted  up  to  999  billion. 

276.  When  more  than  three  figures  are  used  to 
write  a  whole  number,  they  are  separated  by  com- 
mas into  groups,  or  Periods,  of  three  figures  each. 


NUMBERS  ABOVE   10000.  133 

277.  Counting  I'roni  the  right, 

In  the  lirst  j)eriod  any  number  of  units  from  1  to 
999  is  found. 

In  the  second  period  any  number  of  thousands 
from  1  to  999  is  found. 

In  the  third  period  any  number  of  millions  from 
1  to  999  is  found. 

In  tlie  fourth  period  any  number  of  billions  from 
1  to  999  is  found. 


278. 

TABLE. 

m 

(D 

1 

g 

m 

3 

^ 

B       m 

O 

a 

■■?     § 

a    § 

5 

5J          OQ 

OD 

'^  -a    ^ 

-S        S          £ 

i 

o       g 

'S 

mdr 
n-bi 
lion 

indr 
Q-m: 
llior 

s 

T3 

0 

§     i 

=     «     •« 

=     <u     :r 

3 

5      .G 

S 

S     13 

W     Eh     pq 

W     H     ^ 

w 

H     H 

W 

H     t) 

4  2    7 

3   2    0 

4 

0   7 

0 

6   3 

Billions. 

Millions. 

Thousands. 

"" 

Units. 

4th  period. 

3d  period. 

2d  period. 

1st  period. 

279.  427,320,407,063  is  read  four  hundred  twenty - 
seven  billion  three  hundred  twenty  million  four 
hundred  seven  thousand  sixty-three. 

280.  To  read  a  number, 

/.  Beginning  at  the  right,  separate  the  number 
into  periods  of  three  figures  each. 

2.  Beginning  at  the  left,  read  the  number  in 
each  period  and  give  it  the  name  of  the  period, 
except  in  units,  where  the  name  of  the  period  is 
omitted. 


134   WHOLE   NUMBERS  AND  FRACTIONAL  PARTS. 


281.  Oral  Exercise. 
Kead  : 


/. 

10,000. 

9. 

300,000. 

17. 

2,725,248. 

2. 

20,000. 

10. 

401,000. 

18. 

46,701,000. 

3. 

25,000. 

11. 

520,000. 

19. 

530,800,500. 

4. 

38,000. 

12. 

725,001. 

20. 

2,486,430,002. 

5. 

47,001. 

13. 

706,020. 

21. 

8,000,700,049. 

6. 

48,026. 

14. 

.  804,802. 

22. 

3,007,428,304. 

7. 

74,800. 

15. 

900,027. 

23. 

48,006,928,040. 

8. 

59,704. 

16. 

450,006. 

24. 

867,049,302,742, 

282.  Exercise. 

Write  in  words  : 

/.  60,000.  6.  2,425,070.  //.  200,000,000. 

2.  22,700.  7.  5,000,607.  12.  607,007,009. 

3.  48,001.  8.  16,742,003.  13.  4,863,002,140. 

4.  760,040.  9.  24,683,214.  14.  25,600,750,689. 

5.  400,797.  10.  10,020,040.  15.  440,700,632,156. 

283.  Oral  Exercise. 

What  number  is : 

/.  1  greater  than  9,999  ?     1  less  than  10,000  ? 

2.  1  greater  than  59,999  ?     1  less  tlian  40,000  ? 

3.  1  greater  than  599,999  ?     1  less  than  200,000  ? 

4.  1  greater  than  999,999  ?     1  less  than  4,000,000  ? 

5.  In   427,635   there   are   how   many  thousands  ?  how 
many  units  ? 

6.  In  50,700,040  there  are  how  many  millions  ?  how 
many  thousands  ?  how  many  units  ? 

7.  What  is  the  largest  number  that  can  be  written  with 
six  fijjures?  the  smallest  ? 


NUMBERS  ABOVE   10000.  135 

284.  Exercise. 

Write  in  figures  : 
/.  Ten  thousand  one  hundred. 

2.  Fourteen  thousand  two  hundred  six. 

3.  Twenty  thousand  seven  hundred  twenty. 

4.  Fifty  thousand  forty-four. 

5.  Ninety-six  thousand  four  hundred  forty-four. 

6.  One  hundred  thousand  one. 

7.  One  hundred  thousand  one  hundred. 

8.  Two  hundred  thousand  forty-eight. 

9.  Four  hundred  eight  thousand  eighty. 

10.  Five  hundred  thousand  six  hundred  thirty-four. 
//.  Nine  million  seventy  thousand  six  hundred  fifty. 

12.  Twenty-six  million  eight  hundred  thousand  fifty-six. 

13.  Five  hundred  twenty  million  seventy-five  thousand 
fourteen. 

14.  Eight  hundred  million  nine  thousand  five  hundred  six. 

15.  Two  hundred  twenty  million  six  hundred  twenty- 
five  thousand. 

16.  One  billion  one  hundred  million. 

17.  Six  billion  six  million  six  hundred  thousand  four. 

18.  Seventy  billion  sixty-five  thousand  nineteen. 

19.  Nineteen  hundred  five. 

20.  Eighteen  hundred  sixty-five. 

21.  Fourteen  hundred  ninety-two. 

22.  Seventeen  hundred  seventy-six. 

23.  One  hundred  seventy-five  billion  three  hundred  forty- 
eight  million  three  hundred  fourteen  thousand  two  hun- 
dred seventy-seven. 


136    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 

285.  Oral  Exercise. 

/.  What  is  the  largest  number  that  can  be  written  with 
nine  figures  ?  the  smallest  ? 

2.  In  7,777,777  what  does  each  7  represent? 

3.  Head  2,400  as  hundreds  ;  as  tens. 

4.  Head  1,200,000  as  thousands  ;  as  hundreds;  as  tens. 

5.  Give  the  name  of  the  fourth  period  ;  the  first ;  the 
third  ;  the  second. 

6.  Give  the  number  of  the  period  in  which  ten-millions 
are  found  ;  hundred-thousands  ;  hundreds. 

7.  How  many  ciphers  are  required  to  write  one  thou- 
sand ?  one  million  ?  one  hundred  tliousand  ?  one  billion  ? 
ten  thousand  ?  ten  million  ? 


286.  Exercise. 

Add: 

2,991 

2.  1,376 

1,208 

3,259 

3,927 

13,185 

5,954 

2,584 

2,104 

891 

1,014 

5,115 

2,742 

11,606 

1,631 

1,773 

2,880 

2,585 

6,646 

2,403 

3.  37,718 
38,316 
39,385 
36,425 
32,033 
34,871 
28,646 
35,637 
35,393 
33,220 


FUNDAMENTAL  OPERATIONS.  137 


4.       $8.92 

7.                  12 

10.  $89,220,558.49 

8.70 

196,952 

24,322,375.00 

7.62 

323,485 

4,271,562.00 

7.81 

178,733 

568,664.00 

7.89 

1,227 

2,615,202.88 

9.31 

306,428 

3,041,934.00 

8.65 

291,445 

47,722.00 

8.35 

45,759 

13,468,852.32 

7.54 

135,104 

6,909,608.31 

7.32 

425,102 
8.        424,228 

195.41 

5.     $18.56 

//.  $    402,515.47 

18.19 

15,474,447 

154,093.73 

13.76 

4,799,742 

3,785,926.74 

19.72 

6,891,601 

975,429.33 

17.70 

50,711 

3,062,808.61 

15.75 

5,015,965 

611,534.09 

13.53 

1,599,374 

601,806.54 

14.31 

3,617,497 

70,235.22 

11.38 

547,641 

64,029,115.36 

11.72 

1,478,018 

25,178,552.61 

6.  $916.67 

9.        584,120 

12.  $      792,657.23 

625.00 

15,122,948 

2,859,789.02 

458.33 

457,106,995 

964,528.68 

291.67 

263,457,239 

8,035,946.35 

250.00 

26,493,855 

76,730.18 

208.33 

535,872,240 

38,694,831.27 

166.67 

244,909,123 

6,618,659.50 

150.00 

159,311,527 

157,927.87 

150.00 

39,251,413 

22,073,559.51 

150.00 

99,586,188 

1,092,016.89 

138    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 

13.  From        62,727      16.  1,209,090       19.         29,387.00 
take         24,536  1,150,777                     4,097.65 

14.  From        13,000      17.  162,685.27       20.  6,415,000.00 
take          12,880  64,059.85                    6,075.35 

15.  From  4,221,085      18.  2,032,088.69      21.  1,222,117.55 
take       488,086  9,879.79             1,071,828.75 

287.  Exercisco 

Multiply  ; 

/.      170         6.       5,714  //.  43,982.58        16.      719,258 

18                  8,601  .94                     1,328 

2.  4,018          7.     71,015  12.  25,053.86        17.    $150.78 

1,541                 55,602  800                         550 


3.  1,742         8.  807,017        13.      409,297        18.      $47.23 
121  6,951  4,513  7,230 


4.  $L56         9.     $50.41        14.      $131.25        19.      $409.46 
30  750  25  62 


5.  8,781        10.       2,301        15.  1,020,248       20.  0,046,045 
8,107  74,000  7,564  28,031 


FUNDAMENTAL  OPERATIONS.  139 

288.  Exercise. 

Divide :  Measure  : 

/.  5405  by  23.  17.  15,260  by  35. 

2.  $29.76  by  8.  18.  $80.64  by  13.36. 

3.  $15.12  by  42.  19.  304,992  by  72. 

4.  42,804  by  123.  20.  $1,638.00  by  $3.25. 

5.  $1,045.80  by  84.  21.  165,816  by  329. 

6.  $599.36  by  32.  22.  $224.20  by  $5.90. 

7.  135,564  by  316.  23.  $2,058.75  by  $3.05. 

8.  $9,869.04  by  54,  24.  $180.72  by  $15.06. 

9.  18,972  by  204.  25.  $800.80  by  $20.02. 
10.  62,628  by  307.  26.  $145.44  by  $16.16. 
//.  01,306  by  302.  27.  $4,152  by  $10.38. 

12.  $0,789.00  by  325.  28.  $6,586.24  by  $20.08. 

13.  $6,542.82  by  326.  29.  $26,274.78  by  $324.38.  , 

14.  $65,700.50  by  202.  30.  $28,809  by  $800.25. 

15.  2,428,040  by  808.  31.  $28,317.63  by  $325.49. 

16.  1,522,756  by  1,234.  32.  $180,443.76  by  $678.36. 

289.  Exercise. 

/.  A  laborer  had  $125.75  in  a  savings  bank  and  drew 
out  $9.75  each  week  for  12  weeks ;  how  much  had  he  left 
in  this  bank  ? 

2.  In  the  LVIIIth  Congress  there  were  208  Republicans 
in  the  House  of  Representatives  and  178  Democrats ;  how 
many  were  there  of  both  ?  IIow  many  more  Republicans 
than  Democrats  ? 

3.  At  the  beginning  of  a  trip  a  trolley  car  register 
showed  that  6107  fares  had  been  collected,  and  at  the  end 


140    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 


of  the  trip  that  6298  had  been  collected  ;  how  many  fares 
were  collected  during  the  trip  ? 

4.  A  farmer  bought  a  stove  for  $25.75.  He  gave  in  pay- 
ment his  old  stove  weighing  286  pounds,  for  which  he  was 
paid  $0.01  a  pound,  and  the  balance  in  cash  ;  how  mucli 
cash  did  he  pay  ? 

5.  The  following  table  shows  the  amount  and  the  cost 
of  coffee  imported  to  the  United  States  for  11  months  of 
a  certain  year.     Read  the  table  and  find  the  totals : 


Imported  From 

Pounds. 

Cost. 

Central  America 

43,286,988 

29,298,733 

19,348,490 

57,255,375 

695,972,331 

3,746,170 

18.710,738 

4,489.184 

383,428 

14,587,409 

2,665,725 

1,356,436 

4.132,132 

43,439,470 

441.224 

2,081,120 

556,199 

52,897 

Mexico 

West  Indies 

South   America 

Brazil 

Europe 

East  Indies 

Asia  and  Islands 

Africa,  etc 

Totals 

6.  The  distance  from  New  York  to  Sun  Francisco  is 
3250  miles  and  the  distance  from  San  Francisco  to  Ma- 
nila is  6789  miles;  what  is  the  distance  from  New  York 
to  Manila  by  way  of  San  Francisco  ? 

7.  How  many  $100-bills  would  make  a  million  dollars  ? 

8.  A  farmer's  tobacco  crop  consists  of  1426  pounds  of 
wrappers  and  720  pounds  of  fillers.  He  has  two  offers  for 
it :  the  first,  12  cents  a  pound  for  wrappers  and  2  cents  a 
pound  for  fillers  ;  the  second,  9  cents  a  pound  for  all. 
Which  is  the  better  offer,  and  by  how  much  ? 


FACTORS. 


141 


9.  Find  the  sum  of  all  numbers  greater  than  9996  and 
less  than  10005. 

10.  How  many  times  is  a  million  contained  in  a  billion  ? 
//.  What  must  a  man^s  income  be  per  year  in  order  that 

his  income  in  40  years  shall  amount  to  a  million  dollars  ? 

12.  If  a  boy  saves  5  cents  a  day  from  the  age  of  10  years 
to  the  age  of  21  years,  how  much  does  he  save  in  all, 
counting  365  days  to  a  year  ? 

13.  Find  the  increase  in  population  from  1890  to  1900 
for  each  of  the  ten  largest  cities  in  the  United  States,  as 
follows  : 


Cities. 

Population. 

Increase. 

.  1900. 

1890. 

New  York,  N.  Y 

ChicafiTO    III 

3,437,202 
1,698,575 
1,293,697 

575,238 
560,892 
508,957 
381,768 
352,387 
342,782 
325,902 

2,492,591 
1,099,850 
1,046.964 
451,770 
448,477 
434,439 
261,353 
255,664 
298,997 
296,908 

Philadelphia,  Penn 

St.  Louis,  Mo 

Boston,   Mass 

Baltimore    Md 

Cleveland,   Ohio 

Buffalo,  N.  Y 

San  Francisco,  Cal 

Cincinnati,   Ohio 

Factors, 
290.  Exercise. 

/.  What  two  numbers  multiplied  together  will  produce 
4?  6  ?  5  ?  9  ?  10?  15? 

2.  What  three  numbers  multiplied  together  will  produce 
8  ?  12  ?  28  ?  30  ?  42  ? 


142    WHOLE  NUMBERS  AND  FRACTIONAL  PARTS. 

The  numbers  which,  when  multiplied  together, 
will  i)roduce  a  given  number  are  the  Factors  of  the 
number. 

3.  Name  two  factors  that  will  produce  14  ;  22  ;  33  ;  35; 

C4  ;  72  ;  28  ;  84  ;  06. 

4.  What  three  factors  will  produce  45  ? 

45  =  5x9  =  5x3x3.  Therefore,  the  three  factors  of  45  arc 
5,  3,  and  3. 

5.  What  three  factors  will  produce  18  ?  24  ?  28  ?  30  ? 
36  ?  40  ? 

6.  What  are  the  two  factors  of  5  ?  7  ?  11  ?  13  ? 

A  number  that  has  no  other  factors  than  itself  smd 
one  is  a  Prime  Number. 

7.  Tell  which  of  the  following  numbers  are  prime  : 

2  5         8         11         14         17         21 

3  6         9         12         15         18        23 

4  7       10         13         16         20         25 

When  all  the  factors  of  a  number  are  prime  num- 
bers, they  are  called  the  Prime  Factors  of  the  number. 

8.  Find  the  prime  factors  of  36. 

36  =  4x9  =  2x2x3x3.  Therefore,  the  prime  factors  of 
36  are  2,  2,  3,  and  3. 

Find  the  prime  factors  of : 
9.  15.  13.  60.  17.  24. 

10.  30.  14.  72.         18.  35. 

//.  36.         15.  96.  19.  25. 

12.  4S.  16.  27.         20.   32. 


21. 

40. 

25. 

16. 

22. 

56. 

26. 

28. 

23. 

64. 

27. 

45. 

24. 

84. 

28. 

63. 

DIVISORS.  143 


Divisors. 
291.  Exercise. 

/.  Name  a  number  that  will  divide  4  without  a  re- 
mainder. Name  a  number  that  will  divide  9  without  a 
remainder. 

A  number  that  will  divide  a  given  number  without 
a  1  emainder  is  called  an  Exact  Divisor  of  the  number. 

2.  Name  two  exact  divisors  of  6  ;  8  ;  10  ;  14 ;  15 ;  21 
27  ;  35. 

3.  Name  three  exact  divisors  of  12  ;  16  ;  18  ;  20  ;  24 
30  ;  32  ;  36. 

4.  Name  an  exact  divisor  of  both  6  and  9  ;    8  and  12 
10  and  15. 

An  exact  divisor  of  each  of  two  or  more  numbers 
is  called  a  Common  Divisor  of  these  numbers. 

5.  Name  a  common  divisor  of  6  and  8  ;  10  and  15  ;  14 
and  21. 

6.  Name  two  common  divisors  of  8  and  12  ;  9  and  18  ; 
16  and  20. 

7.  Name  three  common  divisors  of  12  and  18  ;  30  and 
42. 

Name  the  greatest  common  divisor  of  : 

8.  20  and  30.        //.  36  and  48.        14.  18  and  24. 

9.  24  and  36.        12.  21  and  28.        15.  16  and  24. 
10.  30  and  45.        13.  36  and  45.        16.  24  and  32. 


CHAPTER  II. 


FRACTIONS. 

Introductory    Problems  and   Definitions. 

292.  Oral  Exercise. 

/.   Line  AB  is  divided  into 

A — 
how  many  equal  parts  ? 

2.  What  is  each  part  called  ? 

3.  How  many  halves  are  there  in  1  ? 

4.  CD  is  divided  into  how  .  , 

C f- 

many  equal  parts  ?  '  I  ' 

5.  What  is  each  part  called  ? 

6.  What  are  2  parts  called  ? 

7.  What  are  3  parts  called  ?    4  parts  ? 

8.  How  many  fourths  are  there  in  1  ? 

9.  EF  is  divided  into  how  i     i     i 
many  equal  parts  ? 

10.  What  is  each  part  called  ? 

//.   What  are  2  parts  called  ?     3  parts?    4  parts  ?     6 
parts  ? 

12.  How  many  eighths  are  there  in  1  ? 

13.  How   many   halves    are   there   in  2  ?      How   many 
fourths  ? 

14.  How  many  fourths  are   there   in  J  ?     How  many 
eighths  ? 


I  '  I  '  I 


PROBLEMS  AND  DEFINITIONS. 


145 


15.  How    many   halves   are   there  in    1 J  ?     How  many 
fourths  ? 

16.  What  is  J  of  J  ?     J  of  i  ? 

17.  How  many  eighths  are  there  in  ^  ?     in  J  ?     in  |  ? 

Read,  supplying  the  missing  numbers  : 


18.   1  =  ^ 

24. 

I    =S 

30. 

2     =4 

36.  f  =  :, 

19.   1  =  i 

25. 

\    =t 

31. 

H  =  i 

37.  1  =  J 

20.  I  =  ; 

26. 

3    =i 

32. 

1      =4 

38.  1  =  i- 

2'-   l  =  t 

27. 

li  =  t 

33. 

f     =4 

39.   i  of  1  = 

22.   i  =  J 

28. 

l}  =  t 

34. 

1  =.; 

40.   }of^  = 

23.   i  =  J 

29. 

ii  =  i 

35. 

i  =i 

->/.   Jof  1  = 

293.  Oral  Exercise. 

/.  Line   MN   is  divided  I 

M \- 

into  how  many  equal  parts  ?  I 

2.  What  is  each  part  called  ? 

5.  What  are  2  parts  called  ?    3  parts  ? 

4.  How  many  thirds  are  there  in  1  ? 

5.  PQ    is    divided    into 
how  many  equal  parts  ? 

6.  What  is  each  part  called  ? 

7.  What  are  2  parts  called  ?     3  parts  ?     6  parts  ? 

5.  How  many  sixths  are  there  in  1  ? 

9.  RS    is     divided    into  ,    ,    , 

R     '    I    I    M    I    I 


I h 


how  many  equal  parts  ? 

10.  What  is  each  part  called?  What  are  2  parts  called? 
12  parts  ?     3  parts  ? 

/  /.  How  many  thirds  are  there  in  2  ?  How  many  sixths  ? 
10 


146 


FRACTIONS. 


12.  What  is  J  of  J  ?    i  of  J  ? 

13.  How  many  sixths  are  there  in  J  ?  How  many 
twelfths  ? 

14.  How  many  sixths  are  there  in  f  ?  How  many 
twelfths  ? 

/5,  How  many  sixths  are  there  in  J  ?  How  many 
twelfths  ? 

16.  How  many  thirds  are  there  in  IJ  ?  How  many 
sixths  ? 

n.  How  many  thirds  are  there  in  If  ?  How  many 
sixths  ? 

18.  How  many  twelfths  are  there  in  |^  ?     in  J  ? 

Read,  supplying  the  missing  numbers  : 


19. 

1=   ^ 

27. 

n=  i 

35. 

3     =* 

43. 

H=  * 

20. 

1=   t 

28. 

n=  i 

3e. 

H=i 

44. 

iS=i 

21. 

1  =  tV 

29. 

H  =  ij 

37. 

H  =  i 

45. 

V=  i 

22. 

i  =    -6 

30. 

If  =  TV 

38. 

3    =i 

46. 

♦  =  4 

23. 

1=  -c 

31. 

■ft"  —    0 

39. 

3    =j 

47. 

J  =  * 

24. 

i  =  TV 

32. 

1  =  i 

40. 

i  ofi=  . 

48. 

i  =rt 

25. 

t  =  TV 

33. 

A=  4 

41. 

Jofi=  . 

49. 

♦  =i 

26. 

t  =  TV 

34. 

H=  5 

42. 

n  =  i 

SO. 

«  =  * 

294.  One  of  the  equal  parts  into  which  a  unit  is 
divided  is  a  Fractional  Unit. 

Thus,  \,  ^,  ^,  ^,  -jij  are  fractional  units. 

295.  A  number  made  up  of  like  fractional  units  is 
a  Fraction. 

Thus,  I,  g,  5,  6  7  V  *ire  fractions. 


REDUCTION.  147 


296.  When  a  fraction  is  written  in  figures,  the 
number  below  the  line  is  called  the  Denominator  and 
the  number  above  the  line  the  Numerator.  The 
numerator  and  denominator  together  are  called  the 
Terms  of  the  fraction. 

Thus,  in  the  fraction  f,  5  is  the  denominator,  4  the  numerator, 
and  4  and  5  togetlier  are  the  terms. 

297.  A  Proper  Fraction  is  a  fraction  whose  numera- 
tor is  less  than  its  denominator ;  as,  |. 

298.  An  Improper  Fraction  is  a  fraction  whose  nu- 
merator is  equal  to  or  greater  than  its  denominator ; 
as,  J  and  J. 

299.  An  Integer  is  a  whole  number  ;  as,  7. 

300.  A  Mixed  Number  is  a  number  made  up  of  an 
integer  and  a  fraction  ;  as,  2|. 

Reduction. 

301.  - 

I    _L     I     J-    I     X    I     J-    I     ±1    X    |-v--|--7--' 

88^888  8T 

Here  we  see  that 

Therefore, 

Multiplying  or  dividing  both  terms  of  a  fraction 
by  the  same  number  does  not  change  its  value. 


148 

FRACTIONS. 

303.  Oral  Exercise. 

, 

Bead,  siipp 

lying  the  missing 

numbers  : 

'■i  =  i 

7.    f  =  5V 

13.   ^  =  i 

'»•  n  =  i 

^-  i  =  i 

«•  J  =  tV 

u.  I  =  t 

'0-  U  =  i 

3.  i  =  i 

9-  t  =  rt 

^5-  J  =  i 

2'-  «  =  i 

^■i  =  i 

'0-    1  -  TT 

^6-  U  =  i 

22-  H  =  i 

«■   *  =  IS 

"•    l  =  5S 

^7.  H  =  t 

'3.  H  =  i 

e-  1  =  51 

'2-    A  =  bV 

/5.  il  =  ^ 

2^-   M  =  * 

303.  Exercise.            , 

1            1 

1                                 1 

Show  tliat : 

1 
mi                                 1 

'               1 
,             1 

1                                 1 
1                                 1 

/.  2J  =  |. 

Thus,     ' 
1 

1             1 

1 

1 ___J 

2.  2  =  |. 

4.  2J  =  f . 

6. 

n  =  Y- 

8.  3f  =  V^. 

5.  3=.|. 

5.  2i  =  i. 

7. 

n  =  V- 

9-  H  =  V- 

304.  Changing  the  form   of    a  number  without 
changing  its  value  is  called  Reduction. 

305.  Reduce  21  to  thirds. 

Since  1  equals  3  thirds,  2  equals  2  times  3  thirds,  or  6  thirds, 
and  2J  equals  6  thirds  plus  2  thirds,  or  8  thirds. 

306.  Reducing  2  to  J,  or  2|  to  }  is  reducing  a  whole 
or  a  mixed  number  to  an  improper  fraction. 

307.  Oral  Exercise. 
Reduce : 

/.  2  to  halves.       5.  4  to  halves. 
2.  4  to  thirds.        4.  3  to  halves. 

Reduce  to  improper  fractions  : 

7.  2J.  10.  8f.  13.  7J. 

8.  SJ.  //.  3i.  14.  6}. 

9.  2f.  12.   3f.  15.  4J. 


5. 

2  to  fifths. 

6. 

5  to  thirds. 

16. 

4J. 

17. 

12J. 

18. 

131. 

REDUCTION. 


149 


308.  Reduce  12|  to  fifths. 

1        s 

12  = 

12f  = 

5' 

13  X  f ,  or  ^/  . 
¥  +  f ,  or  %^. 

309.  Exercise. 

Reduce  to 
/.  131. 

2.  18|. 

3.  161. 

improper  fractions  : 

8.   GG§.              /5. 

5.   42f              16. 

10.  20|.              /7. 

18|. 
55|. 
361. 

22. 
23. 
24. 

112^. 
118f. 
300f. 

4.   14|. 

//.  44f. 

/5. 

165. 

25. 

299|. 

5.  37|. 

6.  ^2\. 

7.  87J. 

12.   30J. 
/5.  48|. 
14.  2G|. 

19. 
20. 
21. 

331. 
14J.    ' 
67J. 

26. 
27. 
28. 

187}. 
575f 

627J. 

310.  Exercise. 

Show  that  :            i     -    i 
/.   \  =  %\.  Thus,  r-^^ ^— 

1 

1 

~3 

1. 
T 

i     i 

2.  1  =  3. 

3.  J  =  3J. 

^-  J -If. 
5.   V  =  2}. 

6.  ^  =  3i, 

7.  ^^=^. 

1 

8. 
9. 

9?  =  i|. 

311.  Changing  |  to  2,  or  |  to  3J,  is  reducing  an 
improper  fraction  to  a  loliole  or  mixed  number. 


iof  2 


312. 


Here  we  see  that    f  =  J  of  2,  or  2  h-  3.     Hence, 
A  fraction  may  he  considered  to  denote  that  the 
numerator  is  to  he  divided  hy  the  denominator. 


150  FRACTIONS. 


313.  Oral  Exercise. 

Reduce  to  a  whole  or  mixed  number  : 
/.    J.      Thus,  I  equals  ^  of  7,  or  2^. 

2.  f.  4.  |.  6.  ^.         8.  Y-  10.  V.  12-  V- 

5.   |.  5.  Y-        7.   y.  P.  J^.  //.  J^.  13.  Y. 

314.  A  fraction  is  in  its  lowest  terms  when  its 
numerator  and  denominator  have  no  common  di- 
visor. 

Thus,  f  is  in  its  lowest  terms,  since  3  and  4  have  no  common 
divisor. 

316.  Reduce  |^  to  lowest  terms : 

Therefore,  to  reduce  a  fraction  to  its  lowest 
terms, 

Divide  out  of  both  terms  all  common  dimsors  of 
them;  or^ 

Divide  hotli  terms  by  their  greatest  common 
divisor. 


316.  Oral  Exercise. 

Reduce  to  lowest  terms  : 

'•  A- 

5.    f,. 

9. 

fS- 

13. 

if- 

17. 

*i- 

2.  «. 

«■  U- 

10. 

«• 

14. 

fJ- 

18. 

«• 

3-  !i 

7-   e- 

11. 

If- 

15. 

fl- 

19. 

n- 

4.  if. 

«■   !«• 

12. 

if- 

16. 

if 

20. 

tW- 

MULTIPLICATION  AND  DIVISION. 


151 


317. 


Multiplication  and   Division. 

4 

'      ^1      I      I     -I--,— 


T—r 


Here  we  see  that 


Therefore, 

To  multiply  a  fraction  hy  a  whole  number^  mul- 
tiply the  numerator  or  divide  the  denominator  by 
the  whole  nuuiber. 

Note.     Divide  the  denominator  when  possible. 

318.  Oral  Exercise. 

Read,  supplying  the  missing  numbers  : 


/.  2  X    1  =  < 

13. 

4 

X 

tV=  • 

2.  2  X    i  =  . 

14. 

3 

X 

1  =  • 

5.   3  X     f  = 

15. 

3 

X 

1  =  • 

4.  2  X  21  = 

16. 

4 

X 

24  =  . 

5.  2  X  4i  =  . 

17. 

5 

X 

5A=  • 

tf.  3  X  9^  = 

18. 

5 

X 

3tV  =   • 

7.  2  X    1  =  . 

19. 

6 

X 

i   -   • 

5.  3  X    f  = 

20. 

3 

X 

4  =  • 

i?.  5  X  A  - 

21. 

2 

X 

i  =  • 

10.  2  x5t%=  < 

22. 

G 

X 

ItV  -  • 

//.   3  x3^  =: 

23. 

4 

X 

3f    =   . 

12.    4    X  5y%=: 

24. 

4 

X 

n  =  • 

152 


FRACTIONS. 


319. 


"j— I r 

I 8 ^ ^ 


Here  we  see  that 


I  -  2  =  J  -  Jl^;  or,  J  -  2  =  I  -  ,-f^. 

Therefore, 

To  divide  a  fraction  hy  a  whole  number,  divide 
the  numerator  or  multiply  the  denominator  hy  the 
whole  number. 

Note.     Divide  tlie  numerator  when  possible. 

320.     Oral  Exercise. 

Kead,  supplying  the  missing  numbers : 


/. 

1 

-^  2  =   . 

2. 

1 

-^  3  =  . 

3. 

f 

-f-  3  =  . 

4. 

i 

-^  4  =  . 

5. 

4| 

-^  2  =  . 

6. 

6| 

-^  3  =  . 

7. 

1 

-f-  3  =  . 

8. 

A 

-^  5  =  o 

Note. 


9-    -h    - 

-10  =z  . 

17.  ^  of    5i 

=   . 

10.     jt     - 

-  5    =   . 

/5.   \  of  IGJ 

=   • 

//.   101  - 

-  5   =   . 

'        /5.   1  of    1 

=   . 

12.  laij  - 

-  3  =  . 

20.   J  of     § 

r=     . 

13.    A    ot    f    =  . 

2/.   1  of    5 

=    . 

14.     ^    of    1    =    . 

22.  i  of    J 

—     • 

/5.     i    of    J    = 

.       23.  \  of  ISi 

=    . 

/fi.     i  of  H    = 

•       24.   i  of  10 jV 

—    • 

i  of  *  hav 

e  the  sail 

le  meaning. 

321.  Oral  Exercise — Miscellaneous  Problems. 
/.  Name  at  sight  the  whole  or  mixed  number  equivalent 
to  each  of  the  following  : 


MISCELLANEOUS  PROBLEMS.  163 

2.  What  part  of  a  quart  is  a  pint  ? 

3.  A  half  peck  is  how  many  quarts  ? 

4.  How  many  quarter  pecks  are  there  in  a  half  peck  ? 

5.  How  many  quarts  are  there  in  3  pints  ? 

6.  f  of  a  foot  is  how  many  inches  ? 

7.  How  many  half  inches  are  there  in  a  foot  ? 

8.  How  many  half  feet  are  there  in  a  yard  ? 

9.  What  part  of  a  dime  is  1  cent  ? 

10.  K  woman  used  half  a  gallon  of  milk  for  breakfast 
and  one  fourth  of  a  gallon  for  dinner ;  how  many  quarts 
did  she  use  for  both  meals  ? 

//.  What  will  f  of  a  pound  of  dried  beef  cost  at  32 
ct.  per  pound  ? 

12.  If  4  pounds  of  coffee  cost  If,  what  part  of  a  dollar 
did  1  pound  cost  ?    How  many  cents  ? 

13.  If  3  pen  trays  cost  $lf,  what  part  of  a  dollar  did 
one  cost  ?    How  many  cents  ? 

14.  How  many  yards  of  ribbon  are  needed  to  make  5 
bows,  each  |  of  a  yard  long  ? 

15.  A.  boy  has  four  2J-dollar  gold  pieces  ;  how  much 
money  has  he  ? 

16.  If  a  $5-bill  is  exchanged  for  half  dollars,  how  many 
should  be  received  ? 

17.  K  rod  contains  5|-  yards  ;  how  many  yards  are  there 
in  2  rods  ? 

18.  How  many  yards  are  there  in  3  rods  ? 


154  FRACTIONS. 


Reduction. 

322.  Reduce  f,  |,  and  i  to  12ths. 

12  =  4  X  3,  3  X  4,  or  2  X  G. 

I  =  ii^  =  A- 
i  =  Tx-l  =  ->^- 

b         2x6  '  ^ 

323.  Fractions  that  have  the  same  number  for  a 
denominator  are  said  to  have  a  Common  Denominator. 

324.  Reducing  f,  |,  and  |  tv)  12ths  is  reducing 
fractions  to  a  common  denominator. 

325.  Fractions  that  have  a  common  denominator 
are  called  Similar  Fractions. 

Thus,  A,  i^if,  JAiid  [•},  haviug  a  common  denominator,  are  similar 
fractions. 

326.  Exercise. 

Reduce : 

/.  I  and  f  to  15ths.  7.  f,  |,  and  \  to  12ths. 

2.  I  and  f  to  20ths.  8.  |,  J,  and  ^  to  30ths. 

3.  \  and  |  to  8ths.  5.  J,  f ,  and  \  to  20ths. 

4.  -I  and  |  to  18ths.  10.  |,  i,  and  f  to  24ths. 

5.  3%  and  ^  to  lOOths.  //.  i,  |,  and  ^  to  60ths. 

6.  f  and  ^  to  3Gths.  12.  |,  ^,  and  f  to  72d8. 

327.  The  Least  Common  Denominator  (1.  c.  d.)  of 
two  or  mor(^  fractions  in  theii*  lowest  terms  is  the 
least  number  tliat  their  denominators  will  divide: 

Thus,  the  least  common  denominator  of  J,  5,  and  \  is  12,  for  12 
is  the  least  number  that  3,  4,  and  6  will  divide. 


REDUCTION.  156 


328.  Reduce  J,  f,  and  }  to  similar  fractions  having 
their  least  common  denominator. 

The  1.  c.  d.  of  i,  f,  and  f  is  readily  seen  to  be  12. 
Now,  12  =  6  X  2,  4  X  3,  or  3  X  4. 

1   _  6^  _    6 
■2  —  6x2  ~   ^''^' 

1  =  3"^  =   1%. 

329.  Exercise. 

Reduce  to  similar  fractions  having  their  least  common 
denominator  : 

/,  f  and  f.  4.  i  and  |.  7.  J,  i,  and  i. 

2.  J  and  f.  5.  }  and  |.  S.  f,  f,  and  |. 

5.  I  and  |.  6.  f  and  Vo-  9.  i,  |,  and  |. 

330.  Reduce  j\,  f,  and  /^  to  similar  fractions  having 
their  least  common  denominator. 

When  the  1.  c.  d.  cannot  readily  be  seen,  as  in  the  above 
problem,  it  may  be  found  thus  : 

12  =  2  X  2  X  3.  The  1.  c.  d.  must  have  in  it  the  prime 

g  _  2  X  2  X  2  factors  2,  2,  and  3  in  order  that  12  may 

divide  it ;  it  must  have  in  it  the  addi- 
tional prime  factor  2,  in  order  tliat  8 
may  divide  it,  and  the  additional  prime 
2  X  3  X  5,  or  120.  factor  5  in  order  that  20  may  divide  it. 


20  =  2  X  2  X  5. 
1.  c.  d.  =  2  X  2  X 


Now,  120  -*- 13  =  10  ;  120  -^  8  =  15  ;  120  -i-  20  =  6. 

120  =  10  X  12,  15  X  8,  or  6  X  20. 

-         10x7  _   la 
'12  ~ioVn-  ■»2o- 

K  15  X  5  7  5 


16x8 


—  "120- 


"s%  —   rIT^  —  '^0' 


166  FRACTIONS. 


331.  Exercise. 

Keduce  to  similar  fractions  having  their  least  common 
denominator  : 

1'  i\,  nh  and  ^j..  5.  A,  ^,  and  Jf-  9.  |J,  \l  and  iJ. 

2.  -I,  -i^,  and  }|.  ff.  A,  f  §,  and  A-  ^^-  i  I'ff.  and  .^ 

•?•  ^S^  T%  and  f§.  7.  ^\,  jK,  and  ^'V-  ^^-  iz^  ^V  and  ^g^. 

^-   2\>  ih  and  iJ.  5.  ^\,  A,  and  g^^.  /2.  ^^,  ^^g,  and  H. 

Addition  and  Subtraction. 

332.  Find  the  sum  of  f,  |,  and  f 

1.  c.  d.  =  40. 

_  6J<J  _  J  5 
"8—6x8  —  ^-^ 
8  _  10  X  3  _  ao 
*  —  10  X  4  "~  to* 

4  _  8x4    _  a« 
^  —  8  X  5    ~  *<>• 

Sura      =         ^S  =  li§. 

333.  Exercise. 

Find  the  sum  of  : 
/.  J,  f ,  and  f         5.  I  ^,  and  ^.         9.  |,  }ji,  and  |. 

2.  },  J,  and  ^.       ff.  },  I,  and  A-  ^^-  tV,  H.  and  H 

3.  I  -,%  and  H.     7.  |,  A,  and  i^.  11.  ^\.  j\,  and  h\. 

4.  I,  /«,  and  ^V     a.  T^,  ^V,  and  ^.  12.  ^,  iJ,  and  f f . 

334.  Find  the  sum  of  12J,  16J,  and  18^. 

12i  =  12ft. 
16J  =  16A. 
18f  =  18ft. 
Sum  =  46H  =  47H. 


ADDITION  AND  SUBTRACTION.  157 

335.  Exercise. 

Find  the  sum  of  : 

/.  6J,  2J,  and  3^.  6.  14f ,  13J,  and  10^^. 

2.  7t,  8i  and  6J.  7.  17i,  28^^,  and  ^^\, 

3.  9f,  7J-,  and  8f  5.  28^1^,  38/o,  and  75. 

4.  S\,  ^,  and  9f.  9.  40f,  7f,  and  125f . 

5.  7|,  9^,  and  4}^.  /O.  84^,  75^^  and  96^V 

336.  Find  the  difference  of  ~-^  and  ■^. 

1.  c.  d.  =    60. 

-a.  _  *iiL  _  jji 

15  ~  4X15  —  6  0- 

3    _  ^^3    _  la 
JljL~  6  X  10  ~  6"- 

Difference         =      i^  =  3V 

337.  Exercise. 

Find  the  difference  of  : 

/.    f  andf.  6.  f  and  f.  //.    |  and  f. 

2.    I  and  |.  7.  i|  and  |.  12.    |  and  |. 

5.  i  and  f  5.  ^  and  Jf  •  ^5.  Ji-  and  yV 

4.  ^V  and  |.  9.  ^\  and  ^g.  /^.  ^  and  -^. 

5.  f  and  f.  /(?.  A  and  A-  ^5.  |f  and  i|. 

338.  Find  the  difference  of  18|  and  9f . 

18f  =  ISif  =  17|i         Sua.     i^  cannot  be   taken  from 
9f  =    9i§  =    9if.  i^  ;  hence,  reduce  1  of  the 

Difference     =  H\t  18  to  |8  and  add  |g  to  H. 


158  FRACTIONS. 


339.  Exercise. 

Find  the  difference  of : 

/.     25    and    6i.  6.  100    and  44f .  //.  lllj  and    55J. 

2.  100    and  37}.  7.  70f  and  25^%.  12.  111^  and    77f 

3.  87}  and  62}.  8.  42f  and  14f .  13.  333}  and  107^^. 

4.  66f  and  33}.  9.  75}  and  28/j.  14.  270}  and  112J. 

5.  87J  and  18|.  10.  90|  and  25|.  /5.  ■'20}  and    '76f. 

340.  Exercise — Miscellaneous  Problems. 

/.  A  market  woman  received  $1J  for  butter,  $2J  for 
eggs,  $4}  for  vegetables,  and  $3 J  for  fruit;  how  much  did 
her  produce  amount  to  ? 

2.  37}  yards  of  bunting  were  sold  from  a  piece  contain- 
ing 50  yards  ;  how  much  remained  ? 

5.  To  make  a  mixed  tea  a  merchant  used  12}  pounds  of 
green  tea  and  18J  pounds  of  black  tea  ;  how  many  pounds 
were  there  in  the  mixture  ? 

4.  The  sum  of  two  numbers  is  100  ;  if  one  of  them  is 
18},  what  is  their  difference  ? 

5.  How  much  change  should  a  customer  receive  from  a 
$10-bill,  if  he  buys  groceries  to  the  amount  of  I5J  and  dry 
goods  to  the  amount  of  $3|  ? 

6.  A  telegraph  pole  40  feet  long  was  set  4|  feet  in  the 
ground  ;  how  many  feet  of  it  were  above  ground  ? 

7.  A  farmer  owns  124^^  acres  of  land  ;  if  25}  acres  of 
this  is  woodland  and  the  rest  farm  land,  how  much  farm 
land  has  he  ? 

8.  A  girl  received  a  mark  of  80}  in  arithmetic  and  S6J 
in  geography  ;  what  was  her  average  in  the  two  studies  ? 


MISCELLANEOUS  PROBLEMS.  159 


9.  Supply  the  missing  number  :  18|  +  37|  4-  ?  =  100. 

fO.  3  turkeys  together  weigh  20^^  pounds  ;  if  one  of 
them  weighs  6f  pounds  and  another  8^  pounds,  what  does 
the  third  weigh  ? 

341.  Exercise — Miscellaneous  Problems. 

/.  j\  of  a  dollar  is  equal  to  how  many  dimes  ?  How 
many  cents  ? 

2.  One-half  of  a  dollar  plus  one-fifth  of  a  dollar  is  how 
many  tenths  of  a  dollar  ?     How  many  cents  ? 

3.  25  quarter-dollars  are  equal  to  how  many  dollars  ? 

4.  A  man  gave  ^  of  his  property  to  each  of  his  4  sons 
and  the  rest  of  it  to  his  wife  ;  what  part  of  the  property 
did  she  receive  ? 

5.  If  half  a  yard  of  silk  costs  $f ,  how  much  does  a  yard  cost  ? 

6.  ^  of  a  gallon  plus  J  of  a  gallon  is  how  many  eighths 
of  a  gallon  ?     How  many  pints  ? 

7.  A  lady  who  charged  $1J  a  day  for  board  should 
charge  how  much  for  6  days'  board  ? 

8.  J  of  a  gallon  of  oil  was  taken  from  a  can  containing 
I^  gallons  ;  what  part  of  a  gallon  remained  ? 

9.  What  does  a  boy  gain  on  10  Sunday  newspapers,  if 
he  buys  them  at  2j-  cents  each  and  sells  them  for  5  cents  ? 

10.  I  exchanged  30  half-dollars  for  dollar  bills  ;  how 
many  bills  did  I  receive  ? 

//.  At  J  of  a  cent  a  pound,  what  is  the  value  of  8 
pounds  of  old  iron  ? 

12.  A  teacher  had  20  pounds  of  candy  put  up  in  half- 
pound  boxes  ;  how  many  boxes  were  there  ? 


160  FRACTIONS. 


13.  How  far  will  a  hoop  go  in  turning  4  times,  if  the 
distance  around  it  is  1 J  feet  ? 

14.  A  rod  contains  5J  yards  ;  how  many  yards  are  there 
in  J  of  a  rod  ? 

15.  Reduce  2f  to  an  improper  fraction. 

16.  At  18  cents  a  dozen,  what  will  'Z^  dozen  eggs  cost  ? 
/7.  0  men  hire  a  car  for  $18 J  ;  how  much  should  each 

man  pay  ? 

18.  J  of  a  gallon  of  maple  syrup  will  fdl  how  many 
bottles,  each  holding  J  of  a  gallon  ? 

19.  A  farmer  bought  a  plow  for  $10  and  gave  in  pay- 
ment wheat  worth  $7J,  and  the  balance  in  cash  ;  how 
much  cash  did  he  pay  ? 

20.  Find  the  cost  of  all  : 

1  pair  of  shoes,  $2J. 
1  pair  of  gloves,  $J. 
1  handkerchief,  %\. 

21.  2J  yards  of  ribbon  were  cut  into  pieces  \  of  a  yard 
long  ;  how  many  pieces  were  made  ? 

22.  If  ;[  of  a  pound  of  tea  costs  $g,  how  much  does  a 
pound  cost  ? 

23.  A  lady  put  J  of  a  pound  of  green  tea  into  a  can  con- 
taining ^  of  a  pound  of  black  tea ;  how  much  tea  was 
there  then  in  the  can  ? 

24.  A  man  bought  a  pair  of 'shoes  for  I2|-  and  gave  in 
payment  a  $5-bill ;  how  much  change  should  he  receive  ? 

25.  A  farmer  put  up  10|  pounds  of  butter  in  half-pound 
prints  ;  how  many  prints  had  he  ? 


MULTIPLICATION.  161 

26.  Reduce  \^  to  a  mixed  number. 

27.  If  5  rakes  cost  $f ,  what  part  of  a  dollar  did  each 
cost  ? 

Multiplication. 

342.         J- 

i \ ' 


1 

J_, 4. 


^  of  4  is  I,  or  1^. 


343.  Oral  Exercise. 

What  is  : 

/.  J  of  3  ?  4.  i  of  5  ?          7.  J  of  2  ?         /O.  J  of  C  ? 

2.  J  of  5  ?  5.  ^  of  7  ?          S.  J  of  :)  ?         //.  }  of  7  ? 

5.  J  of  7  ?  5.  i  of  8  ?          P.  J  of  5  ?         /2.   }  of  11  ? 

344.  What  is  f  of  4  ? 

i  of  4  is  J,  and  §  of  4  is  2  times  ^,  or  |,  or  2|. 
Note,    f  of  4  and  §  x  4  have  tiie  same  meaning. 

345.  Oral  Exercise. 

What  is  : 

/.  I  of  5  ?.  ff.  f  of  7  ?         /a  f  of  6  ?         15.  f  of  3  ? 

2.  f  of  7  ?  7.  I  of  G  ?         //.  I  of  3  ?         16.  f  of  8  ? 

5.  I  of  2  ?  5.  A  of  3  ?         12.   I  of  9  ?         /7.  |  of  4  ? 

4.  2J  X  4,  or  f  of  4  ?  13.   3J  x  3  ?       18.  2f  x  2? 

5.  2|  X  2  ?  5.  3^  X  3  ?       /4.  IJ  X  5  ?       /5.   IJ  x  2? 

11 


102 


FRACTIONS. 


346.  Here  we  see  that  §  of  f  =  t«6  =  |A*. 

+-H--f-h-- 


ioH 


2     t  1 


TofT  |ofi 


Therefore, 

7%6  product  of  two  fractions  equals  the  product 
of  their  numerators  divided  by  the  product  of  their 
denominators. 

Note.     ^  oi  t  and  §  x  f  have  the  same  meaning. 
347.  Multiply  f  by  |. 


|xf=|4|  =  ,V. 

348.  Exercise. 

Multiply  : 

/.    i  X  f             5.  i  X  ^. 

5.  A  X  f .            13.    1    X  f 

2.  1  X  }.             ff.  f  X  4. 

10.   1,2  X  i.            /4.   A  X  |. 

5.    t  X  J.             7.  f  X  i 

//.    1    X  A.          15.   H  X  |. 

4.   f  X  i             S.  f  X  |. 

/2.     J   X  |.             /5.   IJ  X  4. 

349.  Multiply  \i  by  f. 

To  reduce  the  product  to  its 

8      15     ^  X  ;^      1x5      5 
9^16~?'x;^-3x3~6* 

lowest  terms,  we  divide  8  and 
16  by  8,  and  15  and  9  by  3,  and 

8         2 

cancel,  that  is,  ei'ossout^  the  num- 

bers divided. 

In  practice  the  work  may  be  done  thus  : 

1 

9 

s 

5 

;?    5 

2 

MULTIPLICATION. 


163 


350.  2f  by  3J. 

3 

3i  X  2i  =  -^  X  -5 

~  5  -  ''^• 

361.    Exercise 

Find  the  products : 

• 

1-    f    X  V. 

9.      f 

X 

1^ 

17.   IJ  X  IJ. 

2-  i  X  |. 

/O.      t 

X 

2|. 

18.  ^  X  3|. 

5.    f    X  f . 

//.      f 

X 

^' 

19.  7J  X  2J. 

4.    1    X  f . 

/2.      1 

X 

li 

20.   3?j  X  3J. 

5-     f    X  |. 

/5.   2J 

X 

|. 

21.  4f  X  51 

e-  H  X  if. 

/4.   3f 

X 

i 

22.   8|  X  7 J. 

7.  If  X  H. 

/5.  7J 

X 

f. 

25.  4}  X  ^. 

«•  A  X  |^ 

16.  4f 

X 

|. 

24.  31  X  6f. 

352.  Multiply  : 

IGGf  by  7. 

Multiply  126  by  5J. 

166§ 

126 

7 

of 

1163  =  7 

X  16G. 

630  =  5  X  126. 

4!  = 

7xi 

94i  =1  of  126. 

1166§  = 

7  X  166f. 

724i  =  5|  X  126. 

353.  Exercise. 

Multiply  : 

/.  216^ 

5.  350| 

5. 

415^ 

1           13.   135 

6 

8 

15 

6i 

2.  335J 

6.    llSf 

10. 

6001 

\           14.   150 

5 

6 

22 

6J 

5.  206f 

7.  216| 

11. 

406j 

1           15.  420 

9 

25 

14 

12J 

4.  418| 

8.  306| 

12. 

324^^         16.  128 

7 

10 

50 

8i 

164 


t 

FRACTIONS. 

n.  235 

19.  220 

21.  422 

23.  200 

H 

n 

lU 

25J 

18.  160 

20.  406 

22.  270 

24.   605 

13J 

18i 

18t 

122§ 

354.  Oral  Exercise — Miscellaneous  Problems. 


WM 

W//A 

4 

rt 

'■^M 

^^pf»m'^ 

^^ 

Read,  supplying  the  missing  numbers 


/. 

i  =  t 

18. 

4  =  1 V 

55. 

i  =  7V 

2. 

i  =  t 

19. 

i  =  iV 

5ff. 

i  =jTr 

3. 

1=^ 

20. 

i  =  TV 

37. 

*  =  ^V 

4. 

i  =  i 

21. 

i  =  TV 

38. 

?=ijV 

5. 

«  =  t 

22. 

4  +  i  =  rV 

39. 

i  =  zV 

6. 

i+4=6 

23. 

4  -  }  =  rV 

40. 

i  +  i  =  s-o- 

7. 

i-4  =  t 

24. 

1  X  }  =  tV 

41. 

i  +  1  =  jV 

8. 

4  +  1  =  ^ 

25. 

I  +  4  =  tV 

42. 

*  +  f  =  fV 

9. 

f-4  =  « 

26. 

1  +  1  =  TV 

43. 

l-i  =  i 

10. 

§  -  4  =  t 

27. 

1  -  J  =  rV 

44. 

}  -  i  =  sV 

11. 

i~i  =  i 

28. 

|-f  =  ^3 

45. 

f  -  i  =  sv 

12. 

i-i  =  t 

29. 

1  -A=  f3 

46. 

« -  i  =  bv 

13. 

i-|  =  i 

30. 

1-tV=tV 

47. 

i-f  =  jj 

14. 

1  -  1  =  c 

31. 

1  -A=  TV 

48. 

1  -/»=  jV 

15. 

1  -  i  =  0 

32. 

l-+i=TV 

49. 

2  0  -  i  =  jV 

16. 

Jofi=    . 

33. 

4of}=  . 

50. 

i  of  i  =  . 

17. 

iof  J=   . 

34. 

iof4=  • 

61. 

iof}=   . 

MISCELLANEOUS  PROBLEMS.  165 

355.  Oral  Exercise — Miscellaneous  Problems. 

/.  When  thread  is  selling  at  3  spools  for  10  cents,  how 
much  is  that  a  spool  ? 

2.  2  pounds  of  pepper  were  put  up  in  8  packages  of 
equal  size  ;  what  part  of  a  pound  did  each  contain  ? 

3.  4  bottles  of  equal  size  together  hold  a  pint  ;  what 
part  of  a  pint  does  each  hold  ?  What  part  of  a  pint  do 
2  of  them  hold  ? 

4.  What  should  f  of  a  yard  of  silk  cost  at  llj  a  yard  ? 

5.  A  piece  of  cloth  1^  yards  long  shrunk  |  of  a  yard ; 
how  long  was  it  then  ? 

6.  A  can  of  lard  weighed  36J  pounds;  if  the  can 
weighed  2J  pounds,  what  was  the  weight  of  the  lard  ? 

7.  How  much  should  a  laborer  be  paid  for  6  days'  work 
at  Hi  a  day? 

8.  When  matches  are  selling  at  3  boxes  for  5  cents,  how 
much  is  that  per  dozen  boxes  ? 

9.  If  IJ  bushels  of  grass  seed  are  sowed  on  6  acres, 
what  part  of  a  bushel  is  sowed  on  1  acre  ? 

10.  Find  the  cost  of  ^  of  a  pound  of  tea  at  40  cents  a 
pound,  and  |  of  a  pound  of  dried  beef  at  32  cents  a 
pound. 

11.  A  lady  ordered  10  blocks  of  ice  cream,  each  con- 
taining J  of  a  pint ;  how  much  did  it  cost  her,  at  40  cents 
a  quart  ? 

12.  A  lot  containing  |  of  an  acre  was  divided  into  3 
lots  of  equal  size,  and  2  of  them  were  sold  ;  what  part  of 
an  acre  was  sold  ? 

13.  Reduce  ^  to  an  integer. 


166  FRACTIONS. 

14.  How  many  bushels  of  corn  is  |  of  a  barrel,  if  one 
barrel  is  2 J  bushels  ? 

15.  A  lady  had  IJ  dozen  eggs  and  used  9  of  them  ; 
how  many  had  she  left  ? 

16.  Name  the  fraction  equal  to  ||  that  has  the  smallest 
numerator  and  denominator  possible. 

17.  If  poultry  lose  about  I  of  their  weight  in  dressing, 
how  much  should  a  9-pound  turkey  weigh  when  dressed  't 

18.  A  milkman  put  J  of  a  gallon  of  cream  into  pint 
jars  and  sold  each  jar  for  12  cents  ;  how  much  did  he 
receive  ? 

Find  the  cost  of  : 

19.  2J  pounds  of  ham  at  18  cents  a  pound. 

20.  IJ  pounds  of  butter  at  24  cents  a  pound. 

21.  3f  yards  of  cloth  at  12  cents  a  yard. 

22.  2 J  bushels  of  onions  at  $1.50  a  bushel. 

23.  4J  dozen  eggs  at  16  cents  a  dozen. 

24.  3  bushels  of  seed  at  $5J  a  bushel. 

25.  5  pair  of  gloves  at  $1J  a  pair. 

26.  10  pounds  of  twine  at  GJ  cents  a  pound. 

27.  2  hundredweight  of  flour  at  $lf  a  hundredweight. 

28.  0  pounds  of  beefsteak  at  12J  cents  a  pound. 

366.  Exercise— Miscellaneous  Problems. 

/.  When  eggs  are  selling  at  24  cents  a  dozen,  how 
much  should  be  paid  for  10}  dozen? 

2.  66  feet  make  4  rods  ;  how  many  feet  make  a  rod? 

3.  Find  the  selling  price  of  10  sheep  at  an  average  price 
of  $5}  a  head. 


MISCELLANEOUS  PROBLEMS.  167 

4.  Find  the  selling  price  of  65|^  bushels  of  wheat  at 
$0.80  per  bushel. 

5.  Allowing  each  of  25  sailors  1}  pounds  of  meat  daily, 
how  much  meat  must  be  supplied  for  them  for  a  cruise  of 
10  days? 

6.  A  barrel  holding  42J  gallons  is  f  full  of  syrup  ;  how 
many  gallons  of  syrup  are  there  in  the  barrel  ? 

7.  What  should  be  paid  for  12^  cords  of  chestnut  wood 
at  %i^  a  cord  ? 

8.  A  dealer  bought  28  pigs  at  IGJ  a  pair,  and  after  keep- 
ing them  10  weeks  at  a  cost  of  $2 J  a  week,  sold  them  for 
$10J  a  pair  ;  find  his  gain. 

9.  Find  the  weight  of  a  bolt  of  dress  flannel  of  40  yards, 
if  each  yard  weighs  -f^  of  a  pound. 

10.  If  }  of  a  number  is  42,  what  is  2J  times  the  num- 
ber? 

//.  A  farmer  sold  332  pounds  of  leaf  tobacco  at  $0.07| 
a  pound  ;  how  much  did  he  receive  for  it? 

12.  A  dealer  sold  a  plow  for  $6J  ;  if  J  of  this  sum  was 
gain,  what  was  the  cost? 

13.  A  laborer  worked  lOJ  hours  a  day  for  the  6  working 
days  of  a  week  ;  if  his  wages  were  I0.12J  an  hour,  how 
much  did  he  earn  ? 

14.  A  farmer  sows  2 J  bushels  of  wheat  to  the  acre  ;  how 
much  does  he  sow  on  a  field  containing  10 J  acres? 

15.  What  do  3 J  dozen  arithmetics  cost,  at  $5.7G  a  dozen? 

16.  When  hay  is  selling  for  |18f  a  ton,  what  should  be 
paid  for  |  of  a  ton? 

17.  From   a  piece  of  cloth   containing  40  yards  there 


168 


FRACTIONS. 


were   sold  2 J  yards,  18f  yards,  and  6  yards, 
value  of  the  remainder  at  IG  cents  a  yard. 


Find  the 


Division  and   Mensuration. 
357.  4  is  I  of  what  number  ? 


-?-of 

[  T  of  the  number 

3  °^       , 

a  number 

1  xo^  the  number 

I"  -^of  the  number 

If  f  of  a  number  is  4, 
^  of  the   number  is  ^  of 


number    is    3    times   — . 


358.  Oral  Exercise. 

/.  If  J  of  a  number  is  3,  |  of  the  number  is  2  times  — , 
or  — . 

2.  If  I  of  a  number  is  2,  what  is  the  number  ? 

3.  9  is  f  of  what  number  ? 

4.  4  is  1^  of  what  number  ? 

5.  10  is  I  of  what  number  ? 

6.  16  is  I  of  what  number? 

7.  If  I  of  the  length  of  a  line  is  8  inches,  \  of  its  length 
is  ;J  of  —  inches,  or  —  inches,  and  |  of  its  length  is  — 
times  —  inches,  or  —  inches. 

8.  5  of  the  weight  of  a  bushel  of  wheat  is  50  pounds; 
what  is  the  weight  of  a  bushel  of  wheat  ? 

9.  A  boy  spelled  correctly  20  words,  which  was  f  of  the 
number  he  was  given;  how  many  was  he  given  ? 

10.  K  farmer  sold  a  cow  for  32  dollars,  which  was  f  of 
her  cost;  what  was  her  cost? 


DIVISION  AND  MENSURATION. 


169 


359. 

(1)  If  3  X  a  number  =  12,  the  number  ==  12  -^  3. 

(2)  If  3  X  a  number  =  I,  the  number  =  f  -=-  3. 

(3)  If  I  X  a  number  =  |,  the  number  =  f  -^  |. 

Now,  if  I  of  a  number  =  f , 

i  of  the  number  =  J  of  J. 
And  I  of  tbe  number  =  3  x  J  of  I,  or  |  of  5. 


That  is,  the  number  =  f  x  J. 
But  the  number  =  f  ^- 1.     [See  (3).] 
Therefore,  f  -r- 1  =  |  x  |. 
J  is  called  the  inverse  of  f . 

Therefore, 

To  divide  by  a  fraction,  multiply  by  the  inverse 
of  the  fraction. 
360.  Divide  f  by  f. 


*-.*  =  |x- 

=  f  =  i-i. 

Divide  4  by  f . 

4^f  =  §  X  4  = 

=  Y  =  6f . 

361.  Exercise. 

Find  the  quotients  : 

1'  t-|. 

5.  2h 

-|. 

17. 

IJ  -  H. 

^-  f  -  f . 

/O.  3  - 

-f 

18. 

3i-lf. 

5.  4  -  |. 

//.  7- 

-f. 

19. 

^^^. 

4.  ^-|. 

/2.  5- 

-f. 

20. 

H  -  n- 

5-  H  -  J- 

/5.  4- 

-14. 

21. 

If  -  5J. 

6.  J-^tV 

14.  9- 

-3J. 

22. 

If  -^  Gf . 

7.  A-H. 

/5.  8- 

^31. 

23. 

12J-6}. 

«■  i-M. 

/5.  4  - 

-7i. 

24. 

2f  -^  14f . 

170                                      FRACTIONS. 

362.  Divide  233^  by  12J. 

12|    233^ 

G           6 

75)      1400      (18*            Multiply 

both  dividend  and  divisor  by 

75                         the  1.  c.  d. 

of  the  fractions  before  divid- 

em)                  ing. 

600 

M=« 

363.  Exercise. 

Divide  : 

/.  100  by  14f. 

6.  144f  by  6f . 

2.  m\  by  62J. 

7.  262J  by  18J. 

3.  133?5by66f. 

8.  183J  by  26f. 

4.  \n\  by  Gi. 

9.  237i  by  33i. 

5.  187J  by  25. 

10.  666|  by  55|. 

364.  Oral  Exercise. 

/.  If  9  cents  is  ^  of  a  sum  of  money,  what  is  the  sum  ? 

2.  If  J  of  a  melon  costs  10  cents,  what  will  2  melons 
cost  ? 

3.  If  3  pecks  of  potatoes  cost  60  cents,  1  peck  costs  \  of 

—  cents,  or  —  cents;  and  4  pecks  cost  4  x  —  cents,  or  — 
cents. 

4.  If  3  pecks  of  apples  cost  -J  of  a  dollar,  1  peck  costs  \ 
of  —  of  a  dollar,  or  —  of  a  dollar;  and  4  pecks  cost  4  times 

—  of  a  dollar,  or  —  of  a  dollar. 

5.  If  J  of  a  bushel  of  apples  costs  §  of  a  dollar,  }  of  a 
bushel  costs  J  of  —  of  a  dollar,  or  —  of  a  dollar;  and  J  of 
a  bushel  costs  4  times  —  of  a  dollar,  or  —  of  a  dollar. 


DIVISION  AND  MENSURATION.  171 


6.  When  I  of  a  pound  of  tea  cost  36  cents,  how  much 
is  that  a  pound  ? 

7.  A  farmer  received  $24  for  f  of  his  tobacco  crop; 
what  should  he  receive  for  the  whole  crop  at  the  same 
rate?  • 

8.  If  ^  of  a  number  is  1|,  what  is  \  of  the  number  ? 

9.  If  f  of  an  inch  on  a  map  represents  a  mile,  what  does 
an  inch  on  the  map  represent  ? 

10.  3|  is  f  of  what  number? 

11.  K  laborer  saves  $J  a  day,  which  is  f  of  his  daily- 
wages  ;  what  are  his  daily  wages  ? 

12.  2 J  is  f  of  what  number  ? 

13.  If  I  of  the  length  of  a  line  is  2J-  inches,  what  is  J  of 
its  length  ?  What  is  its  length  ? 

14.  In  f  of  a  week  a  laborer  earned  $4.50;  what  should 
he  earn  in  a  week  ? 

/5.  2  dimes  are  f  of  how  many  cents? 

365.  Exercise. 

/.  If  2,000  pounds  of  fertilizer  are  put  up  in  12  bags  of 
equal  size,  how  many  pounds  does  each  bag  contain  ? 

2.  A  farmer  sold  a  load  of  50  bushels  of  oats  which 
weighed  1,575  pounds;  what  did  each  bushel  weigh? 

3.  A  drover  bought  20  sheep  for  $130;  what  was  the 
average  price  paid  per  head  ? 

4.  If  an  echo  traveled  2, 191 J  feet  in  2  seconds,  how  far 
did  it  travel  per  second  ? 

5.  If  it  cost  $136,000  to  build  ^  miles  of  a  trolley  line, 
how  much  did  it  cost  per  mile  on  an  average? 


172  FRACTIONS. 


6.  A  train  that  ran  160  miles  in  2^  hours,  ran  how  far 
each  hour  on  an  average  ? 

7.  How  much  was  paid  per  dozen  for  eggs,  when  $16J 
was  paid  for  54  dozen  ? 

8.  A  can  pay  only  |  of  what  he  owes  B;  if  he  can  pay 
B  $1,004J^,  how  much  does  he  owe  B  ? 

9.  A's  farmer  returned  him  $220J^  from  a  sale  of  corn; 
if  A  received  |  of  the  selling  price  of  the  corn,  how  much 
did  the  farmer  receive  ?     What  did  the  sale  amount  to  ? 

10.  A  bushel  of  corn  weighs  5G  pounds,  which  is  \^  of 
the  weight  of  a  bushel  of  clover  seed ;  what  is  the  weight 
of  a  bushel  of  clover  seed  ? 

//.  A  woodchopper  was  paid  $6J  for  chopping  12 J  cords 
of  wood;  how  much  was  that  a  cord  ? 

12.  A  laborer  earned  $7 J  in  4J  days;  how  much  did  he 
earn  per  day  ? 

366.  How  many  times  is  f  contained  in  J  ? 

-:^  or  1^  X    ^ 


Since  |  equals  A  and  |  equals  \\,  |  is  contained   in  f  as  many 
times  as  8  is  contained  in  9,  or  |,  or  H. 

367.  Oral  Exercise. 

(Reduce  wliole  or  mixed  numbers  to  improper  fractions.) 
How  many  times  is  : 

/.    f  contained  in  |  ?  5.    J  contained  in  1  ? 

2.    J  contained  in  J  ?        4.    f  contained  in  1  ? 


DIVISION  AND  MENSURATION. 


173 


5. 

J  contained 

ill 

f  ? 

/5. 

1\  contained  in  1}  ? 

6. 

1  contained 

in 

2  ? 

/4. 

1   contained  in  |  ? 

7. 

f  contained 

in 

3     ^ 

/5. 

If  contained  in  SJ  ? 

8. 

J  contained 

in 

4  ? 

16. 

3  contained  in   1 1  ? 

9. 

1  contained 

in 

1   ? 

17. 

2|  contained  in  IJ  ? 

10. 

1  contained 

in 

5  ? 

18. 

IJ  contained  in  1    ? 

11. 

f  contained 

in 

1   ? 

19. 

2J  contained  in  3|  ? 

12. 

1  contained 

in 

i  ? 

20. 

2  contained  in  If  ? 

368.  To  measure  }  by  f  is  to  find  how  many  times 
I  is  contained  in  |. 

369.  To  measure  one  fraction  by  another^  reduce 
them  to  a  common   denominator  and  divide  the 

numerator  of  the  first  by  the  numerator  of  the 
second. 

Thus,  f  :  f  =  ,%  -0%  =  9  -5-  8  =  I,  or  H. 


370.  Exercise. 

ind  the  value  of  : 

'•  H  :  f 

4.    7i 

\h 

7-  A 

7J. 

2.    -1  :tV 

5-    fl    • 

h 

8.  25 

:5J 

3-  |t:iV 

6.  16J 

If- 

9.  6| 

34 

371.  Oral  Exercise. 

/.  If  a  boy  has  $2  in  quarter-dollars  and  13  in  half- 
dollars,  how  many  pieces  of  money  has  he  ? 
2.  What  part  of  2  quarts  is  1  quart? 
5.  $1  is  \  of  15,  and  13  is  3  times  \  of  $5,  or  —  of  $5. 
4.  What  part  of  12  inches  is  4  inches  ? 


174  FRACTIONS. 


5.  When  coal  is  selling,  for  $6  a  ton,  what  part  of  a  ton 

can  be  bought  for  $2  ? 

6.  A  lady  spent  $3  for  cloth  at  $|  a  yard;  how  many 
yards  did  she  buy  ? 

Since  $3  equal  $\2^  the  number  of  yards  she  bought  at  $\  a  yard 
was  12  divided  by  3,  or  4. 

7.  IIow  many  ropes,  each  §  of  a  foot  long,  can  be  made 
from  a  6-foot  rope  ? 

8.  When  apples  are  selling  at  $J  a  peck,  how  many  pecks 
can  be  bought  for  $f  ? 

9.  If  coal  is  selling  at  $6  a  ton,  what  part  of  a  ton  can 
be  bought  for  SIJ  ? 

Since  $1^  equal  $|  and  $6  equal  $V.  the  part  of  a  ton  that  can 
be  bought  is  3  divided  by  12,  or  i^^,  or  I. 

10.  When  potatoes  are  selling  at  %^  a  bushel,  how  many 
bushels  can  be  bought  for  $5  ? 

372.  Exercise. 

/.  How  many  times  is  1:2J  contained  in  200? 

2.  At  6J  ct.  a  pound,  how  many  pounds  of  sugar  can 
be  bought  for  75  ct.  ? 

3.  If  the  distance  around  a  wheel  is  7J  ft.,  how  many 
times  will  the  wheel  turn  in  going  2,640  ft.  ? 

4.  A  dealer  bought  cloth  at  75  ct.  a  yard  and  sold  it 
at  87J  ct.  a  yard;  his  gain  was  75  ct.  How  many  yards 
of  this  cloth  did  he  sell  ? 

5.  If  the  cost  of  an  article  is  50  ct.  and  the  selling 
price  62 J^  ct.,  what  part  of  the  cost  is  gained  in  the 
sale? 


DIVISION  AND  MENSURATION.  175 

6.  A  barrel  contains  ol|  gal.  and  a  hogshead  63  gal. 
How  many  barrels  are  there  in  a  hogshead  ? 

7.  If  a  house  valued  at  $5,000  rents  for  I33J  a  month, 
in  how  many  months  will  the  rent  amount  to  the  value  of 
the  house  ? 

8.  How  many  trees  are  there  along  a  street  for  a  distance 
of  330  ft.,  if  they  are  placed  IGJ  ft.  apart? 

373.  Exercise — Miscellaneous  Problems. 

/.  Supply  the  missing  number: 

9f  +  37i  +  50  +  ?  +  6}  =  150. 

2.  A  dealer  bought  a  machine  for  $22J  and  sold  it  at  a 
gain  of  f  of  the  cost;  find  the  selling  price. 

3.  How  many  bushels  were  there  in  my  wheat  crop,  if, 
after  selling  |  of  it,  I  had  353  bu.  left? 

4.  6G  ft.  make  4  rods;  how  many  feet  make  2J  rods? 

5.  Which  is  the  larger  fraction  and  by  how  much,  ^^  or  ^? 

6.  A  farmer  raised  270  bu.  of  oats  on  7J  acres;  what 
was  the  yield  per  acre  ? 

7.  If  8  men  can  do  a  piece  of  work  in  10  days,  how  long 
should  it  take  15  men  to  do  it? 

SuG.  1  man  can  do  the  work  in  8  x  10  da.,  or  80  da. 
15  men  can  do  the  work  in  ^V  of  80  da. 

8.  If  6  men  can  do  a  piece  of  work  in  7 J  days,  how 
long  should  it  take  5  men  to  do  it  ? 

9.  1,000  -  12J  =  ? 

10.  I  bought  a  house  and  40  acres  of  land  for  $5,500. 
If  the  house  was  valued  at  $250,  what  was  the  value  of  the 
land  per  acre  ? 


176  FRACTIONS. 


Complex    Fractions. 
374.  Such    expressions  as  ~  '  -^^  -|-»  and  t^ 

are    called    Complex   Fractions.     They  denote    that 
the  numerator  is  to  be  divided  by  the  denominator. 

f 

Thus,  —T~  denotes  that  f  is  to  be  divided  b}^  4. 


376.  Reduce  • 

-J  to  its  simplest  form. 

1 
4 

=  4^1  =  ^--* 

• 

376.  Exercise, 

Reduce  to  sim 

iplest 

form  : 

/. 

ii. 

4 

'4- 

100 

03J- 

2. 

2 

10.    li. 
5 

100 

3. 

3 

10 

19.  m. 

50 

4. 

2 

i 

12.    «§. 
30 

20.  _L. 
* 

5. 

3 

21.  ±. 

6. 

5 
f 

14.   10  . 
10§ 

22.    l+l 

7. 

6 
V 

23.  37i, 

87i 

8. 

1 

if 

16    100 
37J 

?4.  >>n 

CHAPTER    III. 
DECIMALS. 
Introductory  Problems  and  Definitions. 
377.  Oral  Exercise. 


/.  Into  how  many  equal  strips  is  this  square  divided  ? 

2.  What  part  of  the  whole  square  is  each  strip  ? 

3.  Into  how  many  equal  squares  is  the  first  strip  divided  ? 

4.  What  part  of  a  strip  is  a  small  square  ? 


12 


178  DECIMALS. 


5.  How   many   small   squares   would   the  large  square 
make? 

6.  What  part  of  the  large  square  is  a  small  square  ? 

7.  Into   how  many  equal   strips   is  one  of   the   small 
squares  divided  ? 

8.  How  many  small  strips  would  a  large  strip  make? 

9.  How  many  small  strips  would  the  large  square  make  ? 
10.  A  small  strip  is  what  part  of  the  whole  square? 

//.  How  many  tenths  are  there  in  one? 

12.  How  many  hundredths  are  there  in  one  tenth  ? 

13.  How  many  thousandths  are  there  in  one  hundredth  ? 

14.  \Y\mtk^ot-^? 

15.  Whatis^ij^of  tJ^? 

378.  If  anything  is  divided  into  10,  100,  1000,  etc., 
equal  parts,  these  parts  are  called  Decimal  Parts. 

379.  Any  number  of  decimal  parts  of  a  unit  is  a 
Decimal  Fraction,  or  simply  a  Decimal. 


Reading  and  Writing  Decimals. 

380.  Decimal  fractions  may  be  written  without  a 
denominator.  A  figure  written  in  the  first  place  to 
the  right  of  the  decimal  point  expresses  tenths; 
in  the  second  place,  Jiundredtlis  ;  in  the  third  place, 
thousandths  ;  in  the  fourth  place,  ten -thousandths  ; 
in  the  fifth  place,  hundred-thousandths ;  in  the 
sixth  place,  millionths. 


READING  AND  WRITING  DECIMALS. 


179 


381. 

TABLE. 

GO 

^ 

-O 

^ 

a 

GQ              S 

g 

5              S 

33 

QQ 

1 

13 

"2      o 

i  ^  s 

C 

1 

J3 

fl 
g 

1 

CD 

« 

3          T3          -S 
III 

.2 

'O 

s 

xs 

CO        S 

•s 

'2 

3 

S 

g 

o 
.a 

a       ^ 

g 

O 
,£3 

jL            fl          ^ 

'^ 

» 

^ 

&H 

w 

H          t3 

H 

w 

H 

H       S       !§ 

5 

3 

8 

7 

4 

2     3     . 

2 

3 

5 

G      4      8 

Integral  part 

Decimal  part 

382.  Oral  Exercise. 

/.  Name  the  first  place  to  the  right  of  the  decimal  point; 
the  second;  the  fourth;  the  fifth;  the  third;  the  sixth. 

2.  Beginning  at  the  decimal  point,  name  the  decimal 
places  in  their  order. 

3.  Counting  from  the  decimal  point,  what  is  the  num- 
ber of  the  tenths'  place  ?  the  hundred-thousandths'  ?  the 
thousandths'  ?  the  millionths'  ?  the  hundredths'  ?  the  ten- 
thousandths'  ? 

4.  In  the  number  written  in  the  Table,  how  many  tenths 
are  there?  how  many  thousandths?  how  many  hundred- 
thousandths?  how  many  tens?  how  many  hundreds?  how 
many  thousands?  how  many  millionths?  how  many  millions? 

383.  Oral  Exercise. 
Read : 

/.  .01;  .0001;  .1;  .00001;  .001;  .000001. 

2.  .02.;  .003;  .0004;  .5;  .000007;  .000009. 

3.  .006.;  .7;  .09;  .0006;  .000008;  .000007. 

4.  .3.;  .007;  .08;  .0005;  .000002. 


180 


DECIMALS. 


384.  Exercise. 
Write  in  words: 

/.  .01,  also  .1. 

2.  .001,  also  .00001. 

5.  .0001,  also  .000001. 

4.  .2,  also  .002. 


5.  .0007,  also  .08. 

6.  .00009,  also  .006. 

7.  .02,  also  .0004. 

8.  .007,  also  .00005. 


385.  Exercise. 
Write  as  decimals: 

386. 


*•    TTJJ    1  0  0  0  >    TOlT* 

^'    TIHFOJ    lOOOFffTTJ   IIT* 


.335  =  ^  +  tStt  +  T-u^o-TT  =  iVJl-  +  rh%  +  Tiftfn  =  t^o'I.  =  325  tliou- 
sandths.    Hence, 

Read  the  decimal  as  a  whole  number  and  give  it 
the  name  of  the  place  occupied  by  the  right-hand 
figure. 

Tluis,  .0025  is  read  twenty-Jive  ten-thottsandths,  and  7.64  is  read 
seven  and  sixty-four  hundredths. 


387.  Oral  Exercise. 

Read : 

/.  .25.             6.  .201. 

//. 

.30014. 

16. 

.000204. 

2.  .37.             7.  .0028. 

/2. 

.00154. 

17. 

12.5. 

3.  .752.           8.  .0325. 

13. 

.00302. 

18. 

13.07. 

4.  .082.           9.  .7248. 

14. 

.403. 

19. 

28.375. 

5.   .030.         10.  .2004. 

15. 

7.24. 

20. 

36.004. 

READING  AND  WRITING  DECIMALS.  181 

388.  .02}  is  read  hvo  and  one  four  ill  liundredths ;  jiiid 
3.12J  is  read  three  and  twelve  and  one  half  hundredths. 


389.  Oral  Exercise. 

Read: 

/.  .06}. 

4.   .62 J. 

7.  .08 J. 

/O.  6.66|. 

2.   .12 J. 

5.  .87 J. 

8.  .037 J. 

//.  4.18|. 

5.   .37i. 

6.   .16f. 

9.   .334. 

12.  5. 07 J. 

390.  Exercise. 

"Write  in  decimal  form: 
^-  A-  ^-  tAo- 

-^^    Too"*  °-     100  000* 

^-    TTrflTD"'  ^'    TOUOTTOir- 

13.  18  hundredths. 

/4.  15  ten-thousandths. 

15.  25  thousandths. 

16.  607  thousandths. 
/7.  18  millionths. 

18.  185  hundred-thousandths. 
/5.  325  ten-thousandths. 

20.  Twenty-eight  hundredths. 

21.  Six  and  twenty-one  thousandths. 

22.  Two  hundred  four  millionths. 

23.  Twenty  and  seven  tenths. 

24.  Two  hundred  six  thousandths. 

25.  Two  hundred  and  six  thousandths. 
SuG.  Two  hundred  is  an  integer. 


7. 

m- 

10. 

3A. 

8. 

1 0  b  0* 

11. 

^T^o. 

9. 

1000- 

12. 

8toV 

182  DECIMALS. 


26.  Nine  hundred  and  four  thousandths. 

27.  Six  hundred  and  five  hundredths. 

Reduction. 

391.  Oral  Exercise. 

/.  When  both  terms  of  ^^  are  multiplied  by  10,  what  is 

the  result  ? 

2.  Does  A  equal  ^%  ?     Why  ?     [§  301] . 

3.  When  both  terms  of  -^\  are  multiplied  by  100,  what  is 
the  result  ? 

4.  Does  ^  equal  ^Wir  ?    Why? 

5.  When  both  terms  of  y^o  ^^®  multiplied  by  1000,  what 
is  the  result  ? 

6.  Does  i8_  equal  tWA?     Why? 

7.  Does  j\  equal  A?o^A  ?     Why  ? 

8.  By  what  must  both  terms  of  ^^  be  divided  in  order 
tliat  the  result  may  be  ^  ? 

9.  Does  dividing  both  terms  of  -^  by  10  change  the 
value  of  the  fraction  ?     Why  ? 

10.  Does  f^4  equal  1^ ?     Why? 
//.  Does  1%%  equal  A?     Why? 
12.  Does .^oO^/A  equal  t\?     Why? 

392.  ^  =  m  =  ^A=^^<?A  =  WM' 
Therefore,  .8  =  .80  =  .800  =  .8000  =  .80000. 
Hence, 

Annexing  ciphers  to  a  decimal  does  not  change 
its  value. 


REDUCTION. 


183 


Remomng  ciphers  from  the  right  of  a  decimal 
does  not  change  its  lvalue. 


393.  Exercise. 

Reduce: 
/.  .3  to  hundredths. 

2.  .64  to  thousandths. 

3.  .255  to  ten-thousandths.  9 

4.  .20  to  tenths.  10 

5.  .240  to  hundredths.  //. 

6.  .5000  to  tenths.  12. 


7.  .2  to  thousandths. 

8.  .04  to  thousandths. 
.005  to  millionths. 
.200  to  tenths. 
.6000  to  hundredths. 
.25  to  ten-thousandths. 


394.  Reduce  .425  to  a  common  fraction. 

395.  Exercise. 

Reduce  to  common  fractions  or  mixed  numbers: 
/.   .6.  5.  .075.  9.     7.5. 

2.  .08.  6.   .225.  10.     1.08. 

3.  .25.  7.   .450.  //.  12.012. 

4.  .004.  8.  .0025.  12.     6.775. 

396.  Reduce  .16f  to  a  common  fraction. 

1  fig  —  ^'  —  3ji26i  _     6Q    _  jL 


397.  Exercise. 

Reduce  to  common 

fractions : 

/.  .SJ. 

5.  .14f. 

S. 

.87f. 

2.   .6}. 

6.  .331. 

10. 

.SSJ. 

3,  .l^. 

7.  .66|. 

11. 

.28f 

4.   .16f. 

8.  .18J. 

12. 

.42f. 

184  DECIMALS. 


398.  Reduce  J  to  a  decimal. 

i  equals  i  of  3,  which  equals  i  of  30  tenths. 
I  of  30  tenths  is  7  tenths,  and  3  tenths  remaining. 
LLI —       2  tenths  equals  20  hundredths. 
''^^      i  of  20  hundredtlis  is  5  hundredths. 
Therefore,  f  equals  .75. 

Hence,  to  reduce  a  fraction  to  a  decimal, 
Annex  ciphers  to  the  numerator^  divide  by  the 
denominator,  and  point  off  as  many  decimal  places 
in  the  quotient  as  there  are  ciphers  annexed, 

399.  Another  Method  :  |  =  ^-^  =  ^Vo  =  -"^'^ 
Multiply  both  terms  of  f  by  25  to  reduce  the  frac- 
tion to  hundredths. 

400.  Exercise. 

Reduce  to  decimals  : 

/.   J.                5.   f.  9.  /,.  13.  A. 

2.  i.                6.   i.  10.  A.  1^-  A- 

3.  |.                 7.   A.  //.   ^^.  15.  ^. 

4.  f.                 8.  I.  12.   ~f,.  16.  ^V 

401.  Reduce  |  to  hundredths  ;  |  to  thousandths. 

6)5.00  9)8.000 

~~m\  .8881 

402.  Exercise. 

Reduce : 
/.  J  to  tenths.    3.  |  to  hundredths.   5.  j*^  to  thousandths. 
2.  I  to  tenths.    4.  ^^  to  hundredths.  6.    f  to  thousandths. 


ADDITION  AND  SUBTRACTION.  185 

Addition  and  Subtraction. 

403.  Find  the  sum  of  2.7,  25.08,  .075,  and  27.25. 

2.7 

Write  the  numbers  so  that  units  of  the  same  order 
(vD.Uo 

stand  in  the  same  column,  and  add  as  in   addition  of 
.075 

whole    numbers.      Place   the   decimal   point   between 

27  25 

— '- units  and  tenths  in  the  sum. 

55.105 

404.  Exercise. 

Find  the  sum  of  : 

/.  2.  3. 

3.25  1.006  .3048 

.0075  .25  2.96 

18.2  24.07  .278 

.2785  .8  18.075 


4.  .7,  .045,  and  .62. 

5.  3.18,  9.008,  and  12.7. 

6.  .211,  2.55,  .0175,  and  .004. 

7.  20.02,  128.025,  .07,  and  .7. 

8.  9  tenths,  25  hundredths,  and  17  thousandths. 

9.  8  and  7  tenths,  12  thousandths,  and  4  and  6  hun- 
dredths. 

10.  7  tenths,  87  hundredths,  and  450  thousandths. 

//.  16  and  25  ten-thousandths,  7  and  7  thousandths,  and 
25  hundredths. 

12.  105  millionths,  1  and  70  hundred-thousandths,  44 
and  125  thousandths,  and  1  and  7  tenths. 


186  DECIMALS. 


405.  From  12.75  take  7.9.  From  28.7  take  1.245. 
12.75  28.700 

7.9  1.245 

4.65  27.455 

Write  the  numbers  so  that  units  of  the  same  order  stand  in 
the  same  column,  and  subtract  as  in  subtraction  of  whole  num- 
bers. Place  the  decimal  point  between  units  and  tenths  in  the 
difference. 


406.  Exercise. 

/. 

3. 

5. 

7. 

From  8.362 

2.000 

9.2 

4 

take    2.41 

1.875 

1.075 

2.375 

2. 

4. 

6. 

8. 

From  3.50 

3.2 

25.6 

29.01 

take    1.27 

.75 

1.07 

11.909 

Find  the  difference  between: 

9.  .8  and  .3865.  12.  8.75  and  .109. 

10.  2  and  .0375.  13.  1.09  and  .009. 

//.  4.010  and  .099.  U.  27.4  and  .138. 

15.  65  hundredths  and  75  thousandths. 

16.  1  and  9  tenths  and  99  hundredths. 

17.  109  ten-thousandths  and  109  millionths. 

18.  5  tenths  and  499  thousandths. 

Multiplication,    Division,   and    Mensuration. 

407.  .3  X  .4  =  A  X  A  =  Anr  =  .1^. 

408.  Exercise. 

/.  Show  that  3  X  .4  =  1.2. 
2.  Show  that  .3x2  =  .6. 


MULTIPLICATION,  DIVISION,  MENSURATION.    187 

3.  Show  that  .2  x  .3  =  .06. 

4.  Show  that  .3  x  .04  =  .012. 

5.  Show  that  .03  x  .04  =  .0012. 

6.  In  problem  1,  how  many  decimal  places  are  there  in 
the  multiplicand?  How  many  in  the  multiplier?  How 
many  in  the  product? 

7.  In  problem  2,  how  many  decimal  places  are  there  in 
the  multiplicand?     In  the  multiplier?     In  the  product? 

8.  In  problem  3,  how  does  the  number  of  decimal  places 
in  the  multiplicand  and  multiplier  together  compare  with 
the  number  in  the  product  ? 

9.  In  each  of  problems  4,  5,  and  6,  compare  the  number 
of  decimal  places  in  the  product  with  the  number  in  the 
multiplicand  and  multiplier  together. 

10.  Make  a  statement  about  the  number  of  decimal  places 
in  the  product. 

409.  Multiply  6.5  by  .25. 

6.5 

.25 

325  .25  X  6.5  =  t^o'o-  X  fB  =  i^oi  =  1M%  =  1.625. 

130 
1.625 

From  tMs  illustration  we  see  that  to  multiply 
decimals,  we 

Multiply  as  in  whole  numbers,  and  point  off  as 
many  decimal  places  in  the  product  as  there  are  in 
the  multiplicands  and  multiplier  together. 


188 

DECIMALS. 

410.  Multiply 

.075  by  45;  .075  by 

.45;  .000075  by  420. 

.075 

.075 

.000075 

45 

.45 

4^0 

375 

375 

1500 

300 

300 
.03375 

300 

3.375 

.O3150P 

411.  Exercise. 

Multiply:       /. 

.25  by  7. 

//.  54  by  .06. 

2. 

3.25  by  5. 

12.  4.8  by  7.8. 

3. 

.035  by  9. 

13.  .66  by  .06. 

4. 

2.45  by  12. 

14.  .32  by  20. 

5. 

.095  by  15. 

15.  1.75  by  240. 

6. 

.075  by  14. 

16.  .654  by  2500. 

7. 

8.36  by  40. 

17.  .375  by  .0004. 

8. 

.375  by  30. 

18.  100.5  by  .25. 

9. 

.508  by  25. 

19.  3.96  by  .00256. 

10. 

.0086  by  54. 

20.  .879  by  7.56. 

412.  Show  that  :     10  x  .0025  =  .025. 

SuG.     10  X  .0035  =  10  X  Tiftftfff 

100  X  .0025  =  .25. 

1000  X  .0025  =  2.5. 

From  these  illustrations  we  see  that 

Momng  the  decimal  'point  one  place  rightward 
multiplies  hy  10,  moving  it  two  places  rightward 
multiplies  hy  100,  momng  it  three  places  rightward 
multiplies  hy  1000,  etc. 


MULTIPLICATION,  DIVISION,  MENSURATION.    189 


413.  Oral  Exercise. 

Read,  supplying  the  missing  numbers  : 

/.     10  X       .05  = 

10. 

1000  X 

.075  = 

2.     10  X     .025  = 

11. 

1000  X 

.758  = 

3.     10  X  .0075  = 

12. 

1000  X 

.001  = 

4.  100  X  .0025  =  * 

13. 

100  X 

7.5  = 

5.  100  X     2.85  =  . 

14. 

100  X 

.0075  = 

6.  100  X  .0001  =  . 

15. 

1000  X 

.07  = 

7.     10  X       2.5  =   . 

16. 

100  X 

.5  =  . 

8.     10  X     2.56  =  . 

17. 

1000  X 

.7  = 

9.  100  X  .0075  =  . 

18. 

10000  X 

.00002  =  . 

414.  Show  that : 

2.5 -f 

10  = 

.25. 

Sdg.  2.5  ^  10  =  H  -T-  10. 
2.5  -^  100  =  .025. 
2.5  ^  1000  =  .0025. 
From  these  illustrations  we  see  that 
Moving  the  decimal  point  one  place  leftward 
divides  by  10^  moving  it  two  places  leftward  divides 
by  100,  moving  it  three  places  leftward  divides  by 
1000,  etc. 

415.  Oral  Exercise. 

Read,  supplying  the  missing  numbers  : 


/. 

.5- 

-10  =  . 

8. 

.9-^100  =  . 

15. 

7  -^  1000  =  . 

2. 

.05- 

-10  =  . 

9. 

.27-^100  =  . 

16. 

8.4  -^  1000  =  . 

3. 

2.5- 

-10  =  . 

10. 

.08 -^  100=. 

17. 

75.6-^1000  =. 

4. 

.027- 

-10  =  . 

11. 

9.7  -i-  100=-. 

18. 

.01  -7-  1000=. 

5. 

8.5- 

-10  =  . 

12. 

68-^-100  =  . 

19. 

25-^  1000  =  . 

6. 

.001  - 

-10  =  . 

13. 

37  --  100  =  . 

20. 

.7-^1000  =  . 

7. 

5- 

-  10  = . 

14. 

4  -J-  100  =  . 

21. 

127.5^1000  =  . 

190  DECIMALS. 


416.  Find  the  cost  of  375  rails  at  $6.25  per  hundred  (C). 

275  =  2.75  C 
$6.25  =  the  cost  per  C 
2.75 


3125 
4375 
1250 


117.1875,  or  $17.19  =  the  cost  of  275. 
Find  the  cost  of  2400  shingles  at  $8.25  per  thousand  (M). 


2400  = 

=  2.m  M 

$8.25  = 

=  the  cost  per  M 

2.4 

3300 

1650 

$19,800  =  the  cost  of  2400. 

417.  Exercise. 

Find  the  cost  of; 

/.  600  Havana  cigars  at  $25  per  M. 

2.  275  chestnut  posts  at  $6.50  per  0. 

3.  385  shad  at  $22.50  per  C. 

4.  2500  cartridges  at  $6.50  per  M. 

5.  3250  gun  wads  at  $0.45  per  M. 

6.  225  brass-head  nails  at  $0.52  per  M. 

7.  2250  brass  paper  fasteners  at  $0.35  per  C. 

8.  375  paper  bags  at  $1 .25  per  M. 

9.  250  envelopes  at  $2.25  per  M. 


MULTIPLICATION,  DIVISION,  MENSURATION.    191 

10.  1250  tags  at  $0.95  per  M. 

//.  3400  sheets  of  writing  paper  at  $3.75  per  M. 

12.  10  boxes  of   loaded   shells,   each  containing  25,   at 
$1.25  per  C. 

13.  40  packages  of  letter-heads,  each  containing  500,  at 
$1.36  per  M. 

14.  2275  plastering  laths  at  $5.50  per  M. 

418.  .04  -  2  =  rfo  -  ^  =  ifo  =  -03. 


419.  Exercise. 

/.  Show  that  .9-^3  =  .3. 
2.  Show  that  .06  -^  3  =  .02. 
5.  Show  that  .12^4  =  .03. 

4.  Show  that  .006  -^  2  =  .003. 

5.  Show  that  .015  -^  3  =  .005. 

6.  Show  that  .112  -f-  2  =  .056. 

7.  In  problem  1,  how  many  decimal  places  are  there  in 
the  dividend  ?     How  many  in  the  quotient  ? 

8.  In  problems  2  and  3,  how  many  decimal  places  are 
there  in  the  dividend  ?     How  many  in  the  quotient  ? 

9.  In  problems  4,  5,  and  6,  how  many  decimal  places  are 
there  in  the  dividend  ?     How  many  in  the  quotient  ? 

Notice  that  in  each  of  the  first  6  problems  the  divisor 
is  a  whole  number, 

10.  In  the  first  6  problems,  how  do  the  number  of  deci- 
mal places  in  the  dividend  and  in  the  quotient  compare? 


192 


DECIMALS. 


420.  From  the  illustration  in  Art.  419  we  see  that 
^Yhen  the  dlDtsor  is  a  whole  number,  the  number 
of  decimal  places  in  the  quotient  equals  the  number 
in  the  dividend. 


421.  Divide  7.75  by  25. 

Divide  .004  by  25. 

35)  7.75  (.31 

35). 00400  (.00016 

75 
25 
25 

35 

150 

150 

422.  Exercise. 

Divide:       /-  .005  by  5. 

//.   .03  by  6. 

2.  .027  by  9. 

12.  .012  by  8. 

3.  .0425  by  5. 

13.  .015  by  25. 

4.  .375  by  15. 

14.  .004  by  8. 

5.  8.75  by  25. 

15.  .006  by  8. 

6.  4.00  by  25. 

16.  .00252  by  42. 

7.  25.5  by  15. 

17.  .0125  by  25. 

8.  6.4  by  16. 

18.  191.52  by  42. 

9.  .0144  by  12. 

19.  .0375  by  625. 

10.  .039  by  13. 

20.  .025  by  16. 

423.  Divide  .008  by  .04. 

•««^^-«^-  1000 -^100=   4 

^;w^-4^io--«-^- 

lU 

That  is,  .008  -*-  .04  =  .8  +  4. 

From  this  illustration  we  see  that 

Moving  the  decimal  point  rightward  or  leftward 


MULTIPLICATION,  DIVISION,  MENSURATIjN.     193 

the  same  number  of  places  in  hotli  dividend  and 
divisor  does  not  change  the  quotient. 

424.  Divide: 
(1)  .0625  by  .25  ;  (2)  6.25  by  .0025  ;  (3)  6.25  by  2500. 

(1)  .35). 0625  (   =  25)6.25  (25 

50 

125 

125 

(2)  .0025  )  6.25  (   =  25  )  62500  (  2500 

50 
125 
125 
00 

(3)  2500  )  6.25  (   =  25  )  .0625  ( .0025 
50 
125 
125 

From  these  illustrations  we  see  that,  in  division 
of  decimals,  we 

Move  the  decimal  point  rightward  or  leftward  in 
both  dividend  and  divisor  such  a  number  of  places 
as  will  make  the  divisor  a  whole  number  not  ending 
in  ciphers ;  then  divide  and  point  off  in  the  quo- 
tient as  many  decimal  places  as  there  are  in  the 
dividend. 

13 


1 94  DECIMALS. 


425.  Exercise. 

Divide:    /.  .025  by  .5.  //.  .02  by  .002. 

2.  .075  by  .15.  12.  .5  by  .0005. 

3.  .16  by  .04.  13.  .125  by  .025. 

4.  .036  by  .06.  14.  1  by  .08. 

5.  1.44  by  .02.  15.  1.44  by  .036. 
ff.  .048  by  .12.  16.  86.4  by  .24. 
7.  .144  by  .012.  17.  256  by  200. 
5.  .0375  by  .15.  18.  2.25  by  150. 
9.  .1728  by  .036.  19.  .9  by  360. 

/O.  30.75  by  .075.  20.    4840  by  16. 

426.  In  practice  tlie  division  is  seldom  carried  be- 
yond the  fifth  decimal  place  in  the  quotient,  which 
is  usually  expressed  to  the  nearest  ten-thousandth. 

Thus,  if  the  quotient  is  .76452  +  ,  expressed  to  tlie  nearest  ten- 
thousandth  it  is  .7645.  If  the  quotient  is  .76456  +  ,  expressed  to 
tlie  nearest  ten-thousandth  it  is  .7646. 

Note.  The  sign  +  in  the  above  illustration  indicates  that  the 
quotient  expressed  is  not  complete. 

427.  Exercise. 

Express  to  the  nearest  ten-thousandth  the  value  of  : 
/.  2.573  -^6.  ff.    1  -^15. 

2.  25.37 -T- 8.  7.   .8-^  11. 

3.  .075  -^6.  8.    325.5  -^  16.5. 

4.  37.52  -J-  9.  9.  $375,875  -r  15. 

5.  .046  -J-  .06.  10.  $1000  ^  12. 


MISCELLANEOUS  PROBLEMS.  195 

428.  .25:5  =  ^:^  =  j%  :  M^  =  25  -  50  [§334]  = 
.25  --  5.     That  is,  .25  :  .5  =  .25  --  .5.     Hence, 

To  measure  one  decimal  hy  another  divide  the 
first  hy  the  second. 

429.  Find  the  value  of  .455  :  .35. 

.455  :. 35  =  .35). 455  (      =      35)45.5.(1.3 

35 
105 
105 

430.  Exercise. 

Find  the  value  of  : 

/.  .025  :5.  6.  .0036  :  80. 

2.  .049  :.07.  7.  .096  :  480. 

3.  7.5  :.15.  8.  4.75  :2.5. 

4.  .27  :1.8.  9.  1  :.125. 

5.  150  :.015.  10.  660  :  16.5. 

431.  Oral  Exercise. — Miscellaneous  Problems. 

/.  What  is  .01  of  15  ? 

2.  Read,  supplying  the  missing  numbers: 

(a)  1  -T-  .1  =  .  (c)  100  X  .1  =  o 

(b)  1  -^  100  =   .  (d)  .1  X  .1  =  . 

3.  When  $5  is  gained  on  $100,  what  is  gained  on  $1  ? 

4.  $12  is  .01  ©f  what  sum  ? 

5.  How  long  will  it  take  a  man  to  earn  $12.50  at  $1.25 
a  day? 


196  DECIMALS. 

6.  Read,  supplying  the  missing  numbers: 

(a)  1  -  .2  =   .  (c)  .5  -^  5  3=  . 

(b)  2  4-  .08  =   .  (d)   10  X  .5  =  . 

7.  I  bought  4  pounds  of  coffee  at  35  ct.  a  pound ;  how 
much  change  should  I  receive  from  a  $2- bill  ? 

8.  If  .1  inch  in  a  drawing  represents  a  yard,  how  many 
inches  will  represent  10  yd.? 

9.  At  1.5  ct.  per  pound,  how  many  pounds  of  old  iron 
did  a  boy  sell,  if  he  received  30  ct.  for  it  ? 

10.  Read,  supplying  the  missing  numbers: 

(a)  2.5  +  .05  =   .  (c)  1  -  .01  =  . 

(b)  50  ^  100  =  .  (d)  5  4-  .5  =  . 

11.  K  dealer  paid  $25.50  for  5  lambs;  how  much  was 
that  apiece  ? 

12.  Does.l  =  .10?     Why? 

13.  Find  the  cost  of  a  sheet  of  200  two-cent  stamps. 

14.  Does  10  X  .1  =  1?     Why? 

15.  A  man  who  earns  $2.25  a  day  and  spends  II  a  day 
saves  how  much  in  a  week  (G  da.)  ? 

16.  Doesl  -^  10  =:  .01?     Why? 

432.  Exercise — Miscellaneous  Problems. 

/.  A  man's  store  bill  for  June  was  $28.75,  for  July 
$30.68,  and  for  August  $28.75;  what  was  his  store  bill  for 
the  three  months  ? 

2.  The  value  of  the  English  pound  sterling  in  U.  S. 
money  is  $4.8GG5;  how  much  less  than  $5  is  this? 

3.  The  German  mark  is  worth  $0.2385  in  U.  S.  money; 
how  much  less  than  a  quarter-dollar  is  this  ? 


( 


MISCELLANEOUS  PROBLEMS.  197 

4.  Find  the  amount  of  the  following: 

18  lb.  Sugar  @  6  ct."  per  lb I 

16  lb.  Ham  @  18  ct.  per  lb 

6  lb.  Cheese  @  16  ct.  per  lb 

18  yd.  Muslin  @  12  ct.  per  yd 

% 

5.  There  are  5.5  yd.  in  a  rod;  how  many  yards  are  there 
in  a  line  28  rods  long  ? 

6.  A  cubic  foot  of  water  weighs  62.5  lb.,  and  cork  is  .24 
times  as  heavy  as  water;  what  is  the  weight  of  a  cubic  foot 
of  cork  ? 

7.  A  farmer  sold  128  bu.  of  oats,  which  weighed  3904 
lb. ;  how  much  did  these  oats  weigh  to  the  bushel  ? 

8.  What  price  per  hundredweight  did  a  farmer  receive 
for  a  dressed  porker  that  weighed  3.75  hundredweight  and 
sold  for  $24.31  ? 

9.  When  sugar  is  selling  for  $0.05375  per  pound,  how 
many  pounds  can  be  bought  for  $32.25  ? 

10.  A  merchant  had  on  hand  at  the  beginning  of  a  day's 
sales  $60.18.  During  the  day  he  took  in  $79.76,  and  paid 
out  for  produce  $26.10;  how  much  money  had  he  on  hand 
at  the  close  of  the  day  ? 

433.  Exercise — Miscellaneous  Problems. 
/.  Wliat    should    be    charged   for   2200   lb.    of   coal  at 
$0.32  per  100  1b.? 

2.  Potatoes  are  .162  starch;  how  many  pounds  of  starch 
are  there  in  a  bushel  (60  lb.)  of  potatoes? 

3.  .3  +  .075  -  .0075  =  ? 


198  DECIMALS. 


4.  A  yard  is  36  inches,  and  a  meter  is  3.37  inches  longer 
than  a  yard;  how  long  is  a  meter  ? 

5.  From  a  thousand  take  a  thousandth. 

6.  When  $0.00  is  charged  for  the  loan  of  %\  for  1  year, 
what  should  be  charged  for  the  loan  of  $225  for  \\  years  ? 

7.  A  cubic  foot  of  water  weighs  G2.5  lb.,  and.  zinc  is 
7.19  times  as  heavy  as  water;  what  is  the  weight  of  a  cubic 
foot  of  zinc  ? 

8.  7865  plastering  laths  cost  $31.46;  how  much  was 
that  per  M  ? 

SuG.    7865  =  7.865  M. 

9.  A  dealer  bought  7000  cigars  at  $4.75  per  C,  and 
retailed  them  at  4  for  25  ct. ;  how  much  did  he  gain  ? 

10.  Divide  2002  hundred-thousandths  by  2  thousandths, 
and  write  the  result  in  words. 

//.  If  it  costs  $0.08f  to  ship  100  lb.  of  wheat,  what  will 
it  cost  to  ship  1,250  bu.  of  wheat,  each  weighing  60  lb.? 

434.  Exercise — Miscellaneous  Problems. 

/.  If  a  laborer's  wages  are  20  cents  an  hour,  and  he 
works  9  hours  on  Monday,  10  hours  on  Tuesday,  12  hours 
on  Wednesday,  11  hours  on  Thursday,  11  hours  on  Friday, 
and  6  hours  on  Saturday,  how  much  does  he  earn  that 
week  ? 

2.  A  man  was  employed  as  carrier  in  the  rural  free  de- 
livery service  for  a  year,  for  which  he  was  paid  $600.  Tie 
bought  two  horses,  one  for  $125  and  the  other  for  $110» 
and  paid  $15  a  month  for  their  feed  and  other  expenses; 
he  also  bought  a  wagon  for  $75.     Ilis  board  cost  him  $3.50 


MISCELLANEOUS  PROBLEMS. 


199 


a  week  (52  weeks).  How  much  had  he  clear  at  the  end  of 
the  year,  if  he  sold  the  horses  for  $100  each  and  the  wagon 
for  125  ? 

3.  A  father  willed  $1250  to  each  of  his  two  daughters, 
and  twice  as  much  to  each  of  his  three  sons;  to  his  wife  he 
willed  as  much  as  to  a  son  and  a  daughter;  how  much  did 
he  will  to  all  ? 

4.  Fill  in  the  totals  in  the  following: 


Christians. 

Non-Christians. 

Total. 

Europe 

340,320,000 

124,740,000 

3,800,000 

7,240,000 

12,480,000 
170,000 

America 

Australia 

Asia  and  Africa. . 
Total 

641,550,000 

5.  Which  is  nearer  to  a  million  dollars,  and  by  how 
much,  $1,200,000,  or  $889,008  ? 

6.  I  owed  a  man  a  debt  of  $220  and  gave  him  16  5-dollar 
bills;  how  many  10-dollar  bills  must  I  give  him  to  pay  the 
balance  ? 

7.  A  traveler  has  gone  1220  miles  of  a  journey,  which 
is  f  of  the  distance  he  has  yet  to  go;  how  long  is  his 
journey  ? 

435.  Exercise — Miscellaneous  Problems. 

/.  A  miller  bought  46520  lb.  of  wheat  at  $0.80  per 
bushel  (60  lb.);  find  the  cost. 

2.  A  laborer  worked  for  a  merchant  at  $1.25  a  day  for 
205  days.  He  bought  from  the  merchant  a  suit  of  clothes 
for  $14.75,  a  pair  of  shoes  for  $2.75,  and  groceries  to  the 


BOO 


DECIMALS. 


amount  of  $38.05.     The  balance  was  paid  in  cash;   how 
much  cash  was  paid  ? 

3.  How  many  bushels  of  clover  seed  at  $6.75  per  bushel 
can  be  bought  for  $1000? 

4.  A  farmer  bought  6  steers  at  $30  a  head,  and  after 
feeding  them  4  months  at  a  cost  of  $35  a  month,  sold  them 
for  $410;  find  the  average  gain  per  head. 

5.  The  sum  of  two  numbers  is  27000;  if  the  larger  of 
the  two  numbers  is  17538,  what  is  their  difference? 

6.  324  +  4679  +  ?  +  389  4-  64  +  4642  =  16555. 

7.  Find  the  decrease  in  20  years  in  each  of  the  following 
prices  of  farm  implements  : 


Implements. 


Mowers 

Corn  planters,  hand 

Plows,  walking,  steel 

Plows,  shovel 

Pumps,  wooden 

Rakes,  sulky 

Seeders,  2-horse 

Scythes 

Shellers,  corn 

Sleighs 

Stackers,  hay 

Wagons,  farm 

Hay  carriers 

Churns 

Cultivators,  walking,  2-horse. . 

Potato  diggers 

Corn  drills 

Grain  drills 

Horse  hay  forks 

Harrows 

Potato  hillers 

Fanning  mills 

Harvesters,  twine  binders 

Harvesters,  combined 


1900. 


$65.00 

$40.00 

1.25 

1.00 

15.00 

10.67 

4.00 

2.50 

8.00 

4.00 

25.00 

14.00 

35.00 

25.00 

.80 

.60 

6.00 

2.40 

25.00 

18.00 

55.  GO 

40.00 

90.00 

57.00 

10.00 

3.50 

7.30 

5.33 

20.00 

13.00 

20.00 

10.00 

12.00 

8.00 

50.00 

30.00 

:j.5o 

1.00 

15.00 

10.00 

12.00 

8.00 

35.00 

20.00 

325.00 

120.00 

110.00 

65.00 

Decrease. 


MISCELLANEOUS  PROBLEMS.  SOI 

436.  Exercise — Miscellaneous  Problems. 

/.  A  society  rented  a  hall  for  $25  and  paid  $17.50  to 
advertise  a  lecture.  They  sold  128  tickets  at  $1  each,  326 
at  75  cents  each,  and  122  at  50  cents  each.  How  much 
did  they  clear,  if  they  paid  the  lecturer  $150  ? 

2.  From  f  of  the  sum  of  1905  and  995  subtract  twice 
the  difference  between  701  and  599. 

5.  A  congregation  wishes  to  raise  a  fund  of  $10000.  4 
men  gave  $1250  each,  2  men  $250  each,  and  two  collec- 
tions amounted  to  $795  and  $356  respectively;  how  much 
must  yet  be  raised  ? 

4.  If  a  man  pays  5  cents  for  the  use  of  $1  for  1  year, 
how  much  must  he  pay  for  the  use  of  $500  for  3  years  ? 

5.  If  5  cents  is  paid  for  the  use  of  $1,  how  many  dollars 
should  a  man  have  the  use  of  if  he  pays  $20  ? 

6.  A  dealer  bought  6  cans  of  lard  which  weighed  261 
pounds;  the  empty  cans  weighed  9  pounds.  How  many 
pounds  of  lard  were  there  in  each  can,  if  the  cans  were  of 
equal  size  ? 


CHAPTER  IV. 
DENOMINATE  AMOUNTS. 

437.  Denominate  Units  are  units  established  by  cus- 
tom or  law  to  measure  value,  weight,  time,  length, 
surface,  solids,  capacity,  etc. 

Thus,  a  dollar^  a  pound,  a  day,  a  yard,  etc.,  are  denominate 
units. 

United  States  Money. 

438.  The  standard  unit  of  value  in  the  United 
States  is  the  Dollar. 

TABLE. 

10  mills  (m.)  =  1  cent  (ct.). 
10  cents  =  1  dime  (d.). 

10  dimes  =  1  dollar  ($). 

10  dollars        =  1  eagle  (E.). 

The  coins  of  the  United  States,  now  authorized 
by  law,  are : 

Oold.  Silver.  Nickel.  Bronze. 

Double-Eagle,       Dollar,  Five  Cents.      One  Cent. 

Eagle,  Half-Dollar, 

Half-Eagle,  Quarter-Dollar, 

Quarter-Eagle.      Dime. 
Note  1.     Any  number  of  mills  less  than  10  may  be  expressed 


UNITED  STATES  MONEY.  203 

by  writing  the  figure  iu  the  third  place  to  the  right  of  the  decimal 
point. 

Thus,  5  mills  may  be  expressed  $0,005. 

Note  2.  In  the  settlement  of  business  transactions,  it  is  cus- 
tomary to  omit  any  number  of  mills  less  than  5,  and  to  regard  5 
mills  or  more  as  1  cent. 

Note  3.     $4.8665  =  1  pound  (£),  the  unit  of  value  in  English 
money. 
$0,193  =  1  franc,  the  unit  of  value  in  French  money. 
$0.2385  =  1  mark,    the    unit  of   value   in   German 
money. 

439.  Oral  Exercise. 

/.  How  many  mills  are  there  in  3  ct.?    5ct.?    Id.?    $1  ? 

2.  How  many  mills  are  there  in  |  ct?  Jet.?  Jet.? 
li  ct.? 

3.  How     many    cents    are  there  in  3d.?      6d.?      8d.? 

lOd.? 

4.  How  many  cents  are  there  in  $^  ?     %\  ?     IJ  ?     $|  ? 

5.  How  many  cents  are  there  in  $^  ?     $f?     If?     $|-? 

6.  How  many  dollars  are  there  in  200  ct.?  500  ct.? 
100  ct.?     300  ct.? 

7.  How  many  dimes  should  be  given  in  exchange  for  a 
half-dollar? 

8.  How  many  5-cent  pieces  must  be  given  for  a  quarter- 
dollar  ? 

9.  What  decimal  part  of  a  dollar  is  2d.?  7d.?  3ct.? 
5d.? 

10.  K  boy  has  a  dollar  in  5-cent  pieces;  how  many  coins 
has  he  ? 


204  DENOMINATE  AMOUNTS. 

11.  K  mill  is  what  part  of  a  cent? 

12.  A  cent  is  what  part  of  a  dollar  ? 

13.  A  mill  is  what  part  of  a  dollar? 

14.  How   many  $2^  gold  pieces  must  be  given  in  ex- 
change for  an  eagle  ? 

15.  The  gold  coins  mentioned  in  §  438  have  together 
what  value  ? 

16.  The  silver  coins  mentioned  in  §  438  have  together 
what  value  ? 

17.  IIow  many  pieces  of  money  has  a  boy  who  has  |3i  in 
quarter-dollars  and  50  ct.  in  dimes  ? 

440.  PAPER. 

24  sheets     =  1  quire. 
20  quires     =  1  ream. 

2  reams     =  1  bundle. 

5  bundles  =  1  bale. 

441.  COUNTING. 

12  units    =  1  dozen  (doz.). 

12  dozen  =  1  gross  (gro.). 

12  gross    =  1  great  gross  (G.  gro.). 

20  units   —  1  score. 

442.  Oral  Exercise. 

/.  How  many  units  are  there  iu  6  doz.?    J  doz.?    \^ 
doz.?    2J  doz.? 

2.  How  many  units  are  there  in  3  score?     2  gro.? 

3.  How  many  sheets  of  paper  are  there  in  2  quires? 
J  quire  ? 


PAPER  AND  COUNTING.  205 

4.  A  boy  bought  a  gross  of  matches  for  60  ct.,  and  sold 
them  at  1  ct.  a  box;  find  his  gain. 

5.  What  should  be  paid  for  8  eggs  at  24  ct.  per  dozen  ? 

6.  6  eggs  out  of  4  dozen  were  broken;   how  many  re- 
mained ? 

7.  When  lemons  are  selling  at  the  rate  of  3  for  10  ct., 
how  much  is  that  a  dozen  ? 

8.  When  eggs  are  selling  at  24  ct.  per  dozen,  how  much 
should  be  charged  for  a  score  ? 

9.  When  matches  are  selling  at  3  boxes  for  5  ct.,  how 
much  should  be  paid  for  1  dozen  boxes  ? 

443.  Exercise. 

/.  When  matches  are  selling  at  6  boxes  for  25  ct.,  how 
much  are  they  per  gross  ? 

SuG.     A  gross  is  how  many  times  6  ? 

2.  How  many  dozen  crayons  are  there  in  5  boxes,  each 
containing  a  gross? 

3.  10  eggs  weigh  about  1  pound;  find  the  weight  of  20 
doz.  eggs. 

Find  the  gain  on : 

4.  8  doz.  note-books,  bought  at  $1.60  per  doz.  and  sold 
at  15  ct.  each. 

5.  3  doz.  rulers,  bought  at  50  ct.  per  doz.  and  sold  at 
5  ct.  each. 

6.  4  gro.  erasers,  bought  at  13.20  per  gro.  and  sold  at 
5  ct.  each. 

7.  2  gro.  pencils,  bought  at  $3.00  per  gro.  and  sold  at 
2  for  5  ct. 


206  DENOMINATE  AMOUNTS. 


8.  2  reams  water-color  paper,  bought  at  $20  per  ream, 
and  sold  at  5  ct.  per  sheet. 

9.  5  reams  drawing  paper,  bought  at  35  ct.  per  ream 
and  sold  at  3  sheets  for  1  ct. 

10.  1  quire  blotters,  bought  at  60  ct.  and  sold  at  3  ct. 
each. 

//.  2  M  envelopes,  bought  at  $1.30  per  M  and  sold  at 
25  for  5  ct. 

Bills  and  Accounts. 

444.  A  Bill  is  a  written  statement  of  the  names, 
quantities,  and  prices  of  goods  sold  or  services  ren- 
dered, together  with  the  names  of  the  parties  and 
the  date  of  the  transaction. 

445.  An  Account  is  a  record  of  business  transac- 
tions between  two  parties  at  a  given  date. 

446.  The  Debtor  (Dr.)  is  the  party  who  buys  the 
goods  or  receives  the  service. 

447.  The  Creditor  (Cr.)  is  the  party  who  sells  the 
goods  or  renders  the  service. 

448.  A  Debit  (Dr.)  is  an  entry  in  a  bill  or  an  ac- 
count against  the  debtor. 

449.  A  Credit  (Cr.)  is  an  entry  in  a  bill  or  an 
account  in  favor  of  the  debtor. 

450.  An  Item  is  any  separate  debit  or  credit  made 
in  a  bill  or  an  account. 


BILLS  AND  ACCOUNTS. 


207 


451. 


jyruUb^  diAArnX±h/, 


A   EECEIPTED   BILL. 

West  Chester,  Pa.,  |xxm..  20,  1903. 


Bought  of  THOMAS  HO  AG. 


0  ttv.  Ocvffex 
12  W-.  sh\AXvaA/ 


S5(^ 

70(Ji 

5(P 


2 

10 

1 

JfO 

60 

Jf 

10 


452. 


AN    ACCOUNT. 


Falsington,  Pa.,   iuyru/  l^y  1903. 
In  account  loith  C.   D.   BEANS  «fe  CO. 


Dr. 

10  'uxi.   11UA>^i/yv 

15  U>,   J^XAXUXh/     ■ 

20  ^.  CoA^la^ 

Cr. 

15  doa^.  G<ixi/fi.' 
IS  Uv.  Bu/tttA/ 


,80 


$0.24 


12 


16 


92 

75 
00 


60 
50 
40 


18 


67 


10     50 


8      17 


208 


DENOMINATE  AMOUNTS. 


463.  Exercise. 

/.  Extend  the  items  in  the  following  bill  and  find  the 
total : 

Cleveland,  Ohio,  A/ja>tt/vwU/V  15,  1903, 
m/v.   j3>a/vwtuX  TlLaAyti/n/, 

Bought  of  ROSS    MILLER. 


4  t-OcX^/ix  \Zoii'bAyrUx^  $3.20 
1  dxvo/.  ca/rUi^  QxxVdo\raAXv  QjciAA<iyoX^  5.50 
G  25-lL.  lozou^  'P\AA/nAJ^  .06^ 

5  cAxi/tt/^  I^A^H^Y  GAx:i/ni>-«AW/^  ^'50 

2.  Extend  the  items  in  the  following  account  and  find 
the  balance: 

Baltimore,  Md.,  OyjaA^t  1,  1904. 

In  account  with  STEINMAN  &  CO. 


Dr. 

1  (XcUi^oXt/TU/  J^tcVU^ 

1  otoJiViMX/u  a/Kbd  j^tta/vn>Lv 

2  PUyfHAAM/rui/  <!yUMAJ^ 

Cr. 

^  (ma/.  jhuL<L  PcvtcUx>-tXix 


.80 
.75 


00 


MEASURES  OF  TIME.  209 

3.  Sept.  1,  1903,  Harry  Watsoa  presents  to  you  the  fol- 
lowing bill  for  labor:  10  days  carpentering,  ^2.50  per  day; 
8  days  painting,  $3  per  day;  4  days  roofing,  $2.75  per  day; 
3  days  common  labor,  $1.25  per  day.  AVrite  the  bill  and 
find  the  amount  due. 

4.  Oct.  1,  1903,  James  Wilson  bought  of  Hiram  Leonard, 
Pittsburg,  Pa.,  40  bbl.  AVinter  Superfine  Flour  @  $2.80 
per  bbl.;  12  bbl.  Winter  Extra  Flour  @  $3  per  bbl.;  14 
bbl.  Penna.  Roller  Flour  @  $3.25  per  bbl.  At  the  same 
time  Mr.  Leonard  bought  from  James  Wilson  80  bu.  No.  2 
Red  Wheat  @  80  ct.  per  bu. ;  GO  bu.  Corn  @  55  ct.  per  bn. ; 
50  bu.  Oats  @  43  ct.  per  bu.  Make  out  the  account  and 
find  the  balance  due. 

Measures  of  Time. 
454.  There  are  two  principal  units  of  time,  the 
Day  and  the  Year. 

TABLE. 

60  seconds  (sec.)  =  1  minute  (min.). 

60  minutes  —  1  hour  (hr.). 

24  hours  =  1  day  (da.). 

7  days  =  1  week  (wk.). 

52  weeks  =  1  year  (yr.),  nearly. 

365  days  =  1  common  year. 

366  days  =  1  leap  year. 
100  years  =  1  century. 

The  week  consists  of  7  days  of  which  the  names 
are  as  follows :  Sunday^  Monday,  Tuesday,  Wednes- 
day, TJiuTsday^  Friday ,  Saturday, 


910 


DENOMINATE  AMOUNTS. 


The  year  is  divided  into  12  calendar  months,  as 
follows : 


Names. 

Nam  her 
of  Days. 

Naniee. 

Nambcr 
of  Days. 

January 

February 

March 

April 

May 

June 

31 

28  or  29 
31 
30 
31 
30        1 

July 

August 

September 

October 

November 

December 

31 
31 
30 
31 
30 
31 

Note  1.  A  Centennial  Year  is  one  whose  number  is  divisible  hj 
100. 

Thus,  400,  1800,  and  1900  were  centennial  years. 

Note  2.  Centennial  years  whose  number  is  divisible  by  400,  and 
other  years  whose  number  is  divisible  by  4,  are  Leap  Years. 

Thus,  1600,  1896,  and  1904  were  leap  years. 

Note  3.  a.m.  stands  for  forenoon;  p.m.  stands  for  afternoon; 
7.45  P.M.  means  45  minutes  past  7  p.m. 

465.  Oral  Exercise. 

/.  The  winter  months  are  December,  January,  and  Feb- 
ruary; how  many  days  are  there  in  the  winter  months? 

2.  A  family  that  uses  2  qt.  of  milk  a  day  spends  how 
much  for  milk  during  March,  the  cost  of  the  milk  being 
5  ct.  a  quart  ? 

3.  Name  the  7th  day  of  the  week;  the  5th;  the  2d;  the 
1st;  the  4th;  the  3d;  the  Cth. 

4.  Name  the  months  that  have  30  days. 

5.  AVhen  a  pulse  beats  18  times  in  15  seconds,  how  many 
times  does  it  beat  in  a  minute  ? 

6.  A  building  was  erected  in  MDCCCXC;  how  many 
years  ago  was  that  ? 


MEASURES  OF  CAPACITY.  2 1 1 

7.  Name  the  months  that  have  31  days. 

8.  One  coin  is  dated  1880,  another  1903;  how  many 
years  are  there  between  their  dates  ? 

9.  Name  the  11th  month;  the  4th;  the  7th;  the  12th; 
the  8th;  the  3d;  the  6th;  the  1st;  the  9th;  the  2d;  the 
5th;  the  10th. 

10.  Name  the  Roman  letters  that  are  used  to  write  the 
numbers  of  the  present  year. 

11.  At  20  ct.  an  hour,  how  much  will  a  motorman  earn 
from  6  A.M.  to  2  p.m.? 

12.  How  much  rent  was  paid  in  1  yr.  6  mo.  for  a  house 
that  rented  for  $20  a  month  ? 

13.  When  a  hotel  is  charging  50  ct.  a  meal  and  $1  for 
lodging,  how  much  is  it  charging  per  week  at  the  same 
rate? 

Measures  of  Capacity. 

DRY   MEASURE. 

466.  The  standard  unit  of  dry  measure  is  the 
Bushel. 

TABLE. 

2  pints  (pt.)  =  1  quart  (qt.). 

2  quarts  =  1  small  measure. 

8  quarts  =  1  peck  (pk.). 

4  pecks  =  1  bushel  (bu.)  stricken  measure. 

5  pecks  =  1  heaped  bushel,  nearly. 

LIQUID   MEASURE. 

467.  The  standard  unit  of  liquid  measure  is  the 
Gallon. 


212  DENOMINATE  AMOUNTS. 

TABLE. 

4  gills  (gl)=^  1  pint(pt.). 
2  pints  —  1  quart  (qt.). 

4  quarts        =  1  gallon  (gal.). 
31 J  gallons       =  1  barrel  (bbl.). 
2  barrels       =  1  hogshead  (hhd.)  =  03  gal. 

458.  Oral  Exercise. 

/.  How  many  pints  are  there  in  3  qt.? 

2.  How  many  pints  are  there  in  a  gallon? 

3.  How  many  quarts  are  there  in  J  gal.? 

4.  How  many  quarts  are  there  in  3  pk.? 

5.  How  many  quarts  are  there  in  a  bushel  ? 

6.  How  many  quarts  are  there  in  J  pk.? 

7.  When  apples  are  selling  at  10  ct.  a  half  peck,  how 
much  are  they  a  bushel  ? 

8.  A  farmer  bought  a  peck  of  grass  seed  for  $0.60;  how 
much  was  it  a  bushel  ? 

9.  What  should  be  paid  for  3  qt.  of  vinegar  at  IG  ct.  a 
gallon  ? 

10.  How  many  half -pint  blocks  can  be  made  from  a 
gallon  of  ice  cream  ? 

//.  A  family  that  uses  1  qt.  of  milk  each  week  day  and 
2  qt.  on  Sunday,  uses  how  many  gallons  a  week  ? 

12.  A  trucker  put  1  bu.  of  berries  into  2-qt.  boxes;  how 
many  boxes  were  required  ? 

13.  How  many  gill  bottles  are  required  to  hold  a  quart 
of  alcohol  ? 

14.  When  potatoes  are  selling  at  80  ct.  per  bushel,  what 
should  be  charged  for  J  pk.? 


MEASURES  OF  WEIGHT. 


213 


Measures  of  Weight. 

AVOIRDUPOIS   WEIGHT. 

459.  Avoirdupois  weight  is  used  for  weighing  all 
substances  except  those  weighed  by  Troy  weight, 
medicines  in  prescription,  diamonds,  and  pearls. 
The  standard  unit  is  the  Avoirdupois  Pound. 

TABLE. 

16  ounces  (oz.)         =  1  pound  (lb.). 
25  pounds  =  1  quarter  (qr.). 

4  quarters  =  1  hundredweight  (cwt.). 

20  hundredweight  =  1  Ton  (T.). 
Note  1.     1  lb.  avoir.  =  7000  Troy  grains. 

Note  2.  The  long  hundredweight,  112  pounds,  and  the  long 
ton,  2240  pounds,  are  used  in  custom  houses  and  in  some  wholesale 
and  retail  transactions. 

Thus,  coal  in  Pennsylvania  is  bought  and  sold  by  the  long  ton. 

460.  The  following  table  gives  the  weight  of  a  bushel 
of  various  grains,  seeds  and  other  produce  : 

TABLE. 


Lb. 

Lh. 

LI. 

Barley 

48 

Corn  (shelled) 

56 

Potatoes 

60 

Beans 

00 

Corn  (on  cob) 

70 

Rye 

56 

Buckwheat 

48 

Clover  seed 

60 

Timothy  seed 

45 

Bran 

20 

Oats 

32 

Wheat 

60 

Note.     196  lb.  is  the  weight  of  1  barrel  of  flour. 

200  lb.  is  the  weiglit  of  1  barrel  of  beef  or  pork. 
100  lb.  is  the  weight  of  1  kec:  of  nails. 


214  DENOMINATE  AMOUNTS. 


TROY    WEIGHT. 

461.  Troy  weight  is  used  in  weighing  gold,  silver, 
platinum,  and  jewels,  except  precious  stones.  The 
standard  unit  is  the  Troy  Pound. 


TABLE. 

24  grains  (gr.)      =  1  pennyweight  (pwt.). 
20  pennyweights  =  1  ounce  (oz.). 
12  ounces  =  1  pound  (lb.). 

Note  1.  The  unit  of  weight  for  precious  stones  is  the  Carat, 
which  equals  4  diamond  grains,  or  about  3^  Troy  grains.  Pearls 
are  sold  by  the  diamond  grain. 

Note  2.  412.5  gr.  is  the  weight  of  a  silver  dollar. 

462.  Oral  Exercise. 

/.  What  should  4  oz.  of  cheese  cost  at  IG  ct.  per  pound  ? 

2.  How  many  more  pounds  are  there  in  a  long  ton  than 
in  a  short  ton  ? 

3.  What  should  1000  lb.  of  hay  cost  at  118  per  ton  ? 

4.  How  many  200-lb.  sacks  of  fertilizer  will  make  a  ton  ? 

5.  What  part  of  a  pennyweight  is  8  gr.? 

6.  IIow  much  should  \  bu.  of  wheat  weigh  ? 

7.  How  much  should  a  peck  of  potatoes  weigh? 

8.  How  many  bales  of  hay,  each  weighing  4  hundred- 
weight, will  weigh  a  ton  ? 

9.  What  will  J  lb.  of  spices  cost  at  2  ct.  an  ounce  ? 


MEASURES  OF  WEIGHT.  215 

10.  What  will  1  lb.  12  oz.  of  dried  beef  cost  at  24  ct.  per 
pound  ? 

//.  6  pieces  of  silverware,  each  weighing  4  oz.,  weigh 
how  many  pounds  ? 

12.  How  many  grains  of  gold  are  there  in  a  piece  of 
jewelry  weighing  2  pwt.,  if  it  is  f  pure  gold  ? 

13.  Find  the  weight  of  2  bu.  of  oats. 

463.  Find  the  cost  of  4500  lb.  of  hay  at  $18.75  per  ton. 


4500  lb. 

=  4.500  M  lb.  = 

:  2.25  T. 

$18.75  = 

:  the  cost  per  T. 

2.25 

9375 

3750 

3750 

$42.1875,  or  $42.19  =  the  entire  cost. 

464.  Exercise. 

Find  the  cost  of: 
/.  1875  lb.  of  bran  at  $18  per  ton. 

2.  3880  lb.  of  straw  at  $0.25  per  ton. 

3.  4665  lb.  of  fertilizer  at  $14  per  ton. 

4.  1840  lb.  of  hay  at  $18.25  per  ton. 

5.  3250  lb.  of  mixed  feed  at  $20.25  per  ton. 

6.  8560  lb.  of  phosphate  at  $18  per  ton. 

7.  10756  lb.  of  iron  ore  at  $5.78  per  ton. 

8.  278  lb.  of  bran  at  $18.50  per  ton. 

9.  3425  lb.  of  old  iron  at  $20  per  ton. 
10.  4596  lb.  of  coal  at  $5.75  per  ton. 


216 

DENOMINATE  AMOUNTS. 

Measures  of  Length. 
465.  Exercise. 

1       1 

3 

1        1 

1         1 
2 

1         1 

1         i 
1 

'  1  ' 

/.  How  many  inches  long  is  this  rule? 

2.  How  many  half  inches  are  there  in  each  inch? 

3.  How  many  half  inches  are  there  in  2  inches?  in  3 
inches? 

4.  How  many  quarter  inches  are  there  in  each  inch? 

5.  How  many  quarter  inches  are  there  in  2  inches?  in  3 
inches? 

6.  How  many  eighth  inches  are  there  in  each  inch? 

7.  How  many  eighth  inches  are  there  in  2  inches?  in  3 
inches? 

8.  How  many  half  inches  are  there  in  IJ  inches?  in  2 J 
inches? 

9.  How  many  quarter  inches  are  there  in  IJ  inches?  in 
1|  inches? 

10.  Measure  these  lines  and  state  how  many  inches  long 
each  is. 


//.  Draw  a  line  IJ  inches  long. 
12.  Draw  a  line  12  inches  long. 

12  inches  make  a  foot* 


MEASURES  OF  LENGTH.  217 

13.  Draw  a  line  1  foot  6  inches  long.    How  many  inches 
long  is  it? 

14.  How  many  inches  are  there  in  a  half  foot?  In  IJ  ft.? 

15.  What  part  of  a  foot  is  6  inches? 

16.  What  part  of  a  foot  is  4  inches? 

17.  What  part  of  a  foot  is  3  inches? 

18.  Measure  this  line. 

, 1 

How  long  is  it?    How  long  is  each  part  of  it? 

19.  Let  each  part  represent  a  foot ;  how  many  feet  does 
the  whole  line  represent? 

3  feet  make  a  yard. 

20.  What  does  the  line  above  represent? 

21.  How  many  feet  are  there  in  half  a  yard? 

22.  How  many  inches  are  there  in  a  yard? 

23.  How  many  inches  are  there  in  half  a  yard? 

24.  How  many  inches  are  there  in  J  yard?   in  |  yard? 

25.  How  many  inches  are  there  in  |  yard?   in  f  yard? 

26.  What  part  of  a  yard  is  a  foot? 

27.  What  part  of  a  yard  is  2  feet? 

28.  What  part  of  a  yard  is  6  inches? 

29.  What  part  of  a  yard  is  4  inches? 

466.  The  standard  unit  of  length  is  the  Yaed. 

TABLE. 

12  inches  (in.)  =  1  foot  (ft.). 
3  feet  =  1  yard  (yd.). 
5 J  yards  =  1  rod  (rd.). 
320  rods  =  1  statute  mile  (mi.). 


218  DENOMINATE  NUMBERS. 

467.  Other  Measures  of  Length. 

A  Fathom  =  6  feet,  used  to  measure  the  depth  of  the 
sea. 

A  Hand  =  4  inches,  used  in  measuring  a  horse's 
height. 

A  Geographic  Mile,  or  Knot  =  G08G  ft. 

A  League  (England  and  U.  S.)  =  3  geographic 
miles,  used  to  measure  distances  at  sea. 

A  Chain  =  4  rd.,  used  by  surveyors  in  measuring  land. 

468.  Oral  Exercise. 

/.  How  many  inches  long  is  a  2-foot  rule? 

2.  How  many  18-in.  ropes  can  be  made  from  a  rope  6 
ft.  long? 

3.  How  many  yards  are  there  in  3  rd.  ? 

4.  How  many  inches  are  there  in  a  J-ft.  rule? 

5.  The  ocean  at  a  certain  point  is  20  fathoms  deep;  how 
many  feet  is  that? 

6.  A  horse  that  is  14J  hands  high  is  how  many  inches 
high? 

7.  How  many  feet  are  there  in  3G  inches? 

8.  4  inches  is  what  part  of  a  foot? 

9.  What  is  the  distance  around  a  room  24  ft.  long  and 
18  ft.  wide? 

10.  A  rule  1  k  ft.  long  was  marked  off  in  inches;  how 
many  inches  were  marked  off? 

11.  K  boy's  hoop  in  turning  3  times  passes  over  10  ft.: 
what  is  the  distance  around  it? 


MISCELLANEOUS  PROBLEMS.  219 

12.  How  many  feet  are  there  in  a  rod? 

13.  How  many  feet  high  is  a  horse  that  is  15  hands  high? 

469.  Oral  Exercise — Miscellaneous  Problems. 

Read,  supplying  the  missing  numbers  : 
/.  .5  of  18=. 

2.  2480  -f-  8  =  . 

3.  f  of  100  = . 

4.  8  =  1  of. 

5.  ^  +  ^=» 

6.  %l  -  87  ct.  =  —  ct. 

7.  35  ct.  +  •  =  II. 

8.  4000  lb.  =  —  tons. 

9.  28  da.  =  —  weeks. 

10.  IJ  yd.  of  cloth  at  18  ct.  per  yard  cost  —  ct. 

//.  .25  of  244  =  . 

12.  200  shad  at  $20  per  C  cost  — . 

13.  3000  cigars  at  $30  per  M  cost  — . 

14.  3000  lb.  of  hay  at  $18  per  T.  cost  — . 

15.  1.5  ft.  =  —  of  a  yard. 

16.  2  -^  100=. 

17.  $20  —  25  ct.  =. 

470.  Exercise — Miscellaneous  Problems. 

/.  If  chestnuts  are  bought  at  $2.40  a  bushel  and  sold  at 
5  ct.  a  pint,  find  the  gain  on  a  bushel. 

2.  How  many  seconds  are  there  in  an  hour  ? 

3.  When  vinegar  is  retailed  at  3  ct.  per  quart,  how 
much  is  received  for  a  barrel  containing  36  gal.  ? 


220  DENOMINATE  NUMBERS. 

4.  A  dealer  wishes  to  sell  apples  that  cost  him  $1.20  per 
bushel  so  as  to  double  his  money  ;  how  much  must  he 
cliarge  per  peck  ? 

5.  How  many  hours  are  there  in  October  ? 

6.  What  should  be  paid  per  bushel  for  oats  that  weigh 
only  28  lb.  to  the  bushel,  when  oats  are  quoted  at  36  ct. 
per  bushel  (32  lb.)  ? 

7.  In  $7.50,  how  many  dimes  are  there  ?  How  many 
cents  ? 

8.  At  15  ct.  per  cwt.,  find  the  cost  of  shipping  a  ma- 
chine weighing  3250  lb. 

9.  How  many  minute  spaces  does  the  minute  hand  of  a 
clock  pass  over  in  a  day  ? 

10.  How  many  bottles,  each  holding  a  gill,  can  be  filled 
from  a  gallon  of  ink  ? 

11.  K  clock  that  strikes  the  hours,  strikes  how  many 
times  a  day  ? 

12.  A  dealer  put  3  pk.  of  peanuts  into  pint  packages; 
how  many  packages  had  he? 

13.  If  J  bu.  of  oats  weighs  14^  lb.,  how  much  too  light 
are  these  oats  per  bushel  ? 

14.  A  ship  was  sunk  in  80  fathoms  of  water  ;  how  many 
feet  was  that  ? 

15.  Show  that  a  mile  contains  1760  yd. 

16.  Find  the  cost  of  the  fertilizer  needed  for  6  acres  of 
land,  if  400  lb.  are  sowed  to  the  acre,  and  it  costs  $14  per 
ton. 

17.  K  miller  bought  9540  lb.  of  corn  at  60  ct.  a  bushel ; 
how  much  did  he  pay  for  it  ? 


MEASURES  OF  SURFACE.         221 


18.  When  bran  is  selling  for  %'Z4:  per  ton,  how  many 
pounds  should  be  given  for  $15  ? 

19.  4J  bu.  of  wheat  make  a  barrel  of  flour ;  how  many- 
pounds  of  flour  does  1  bu.  of  wheat  make  ? 

20.  How  many  2-grain  quinine  pills  can  be  made  from  a 
pound  (avoir.)  of  quinine  ? 

21.  A  farmer's  wheat  crop  weighs  16515  lb.;  how  much 
is  it  worth  at  80  ct.  a  bushel  ? 

22.  What  should  be  charged  for  a  sack  of  flour  weighing 
24^  lb.,  when  flour  is  selling  at  $4.80  a  barrel  ? 

23.  Siiow  that  a  mile  contains  5280  ft. 

24.  What  years  since  1890  have  been  leap  years  ? 

25.  A  surveyor's  chain  contains  100  links,  each  7.92  in. 
long  ;  how  many  feet  long  is  the  chain  ? 

26.  How  many  days  are  there  from  June  1  to  Sept.  15  ? 
SuG.   Count  Sept.  15  but  not  June  1. 

27.  How  many  days  are  there  from  June  15  to  Dec.  31  ? 

28.  What  date  is  00  da.  after  Apr.  20  ? 

Measures  of  Surface. 
471.  The  opening  between  two  lines  that  meet  is 
called  an  Angle. 


Acute  Angle  Right  Angle  Obtuse  Angle 

472.  Anything  that  has  length  and  breadth  with- 
out thickness  is  called  a  Surface. 


228 


DENOMINATE  AMOUNTS. 


473.  A  Square  is  a  surface  that 
has  four  equal  sides  and  four 
right  angles. 

474.  Exercise. 
/.  "Which  is  larger,  an  acute  angle  or 

A  Square.  a  right  angle  ?     A  right  angle  or  an 

obtuse  angle  ?     An  acute  angle  or  an  obtuse  angle  ? 

2.  Draw  an  acute  angle  ;  an  obtuse  angle  ;  a  right  angle. 

3.  Draw  a  square  2  inches  long. 

4.  Draw  a  square  3  inches  long. 

5.  Draw  a  square  ^  inch  long.  *^ 

476.  Exercise. 

/.  Measure  the  length  and  the  width  of  the  square  above. 

How  long  is  it  ?     How  wide  ? 

A  square  one  inch  long  and  one  inch  wide  is  a  Square 
Inch  {sq.  in.). 

2.  Draw  a  square 
inch. 

5.  H o w  1 0 ng  is 
square  ABCD  ?  How 
wide  ? 

4.  What  is  the  dis- 
tance around  it  ? 

5.  Howmany square 


inches    does    it   con- 
tain ? 

6.  How  many  sides 
has  a  square  ? 


A 


MEASURES  OF  SURFACE. 


223 


7.  How  do  the  sides  compare  in  length  ? 

8.  Draw  a  square  Ifoot  long  and  1  foot  wide. 

A  square  one  foot  long  and  one  foot  wide  is  a  Square 
Foot  {sq.  ft,). 

Here   is  a  drawing  representing  a  square  foot   divided 
into  square  inches. 

9.  How  many  square  inches 
are  there  in  1  row  ? 

10.  How  many  rows  of  square 
inches  are  there  ? 

//.  How  many  square  inches 
are  there  in  a  square  foot? 

12.  How  many  inches  long 
are  the  four  sides  of  a  square 
foot? 

13.  How  long  is  i  of  a  square  foot  ? 

14.  How  many  inches  is  it  around  i  of  a  square  foot  ? 

15.  How  many  square  inches  are  there  in  ^  of  a  square 
foot? 

16.  How  many  square  inches  are  there  in  f  of  a  square 
foot  ? 

/  7.  How  many  square  inches  are  there  in  |  of  a  square  foot  ? 

18.  How  many  square  inches  are  there  in  -j'g-  of  a  square 
foot? 

What  part  of  a  square  foot  is  : 

19.  12  sq.  in.  22.  48  sq.  in. 

20.  24  sq.  in.  23.   GO  sq.  in. 

21.  36  sq.  in.  24.  72  sq.  in. 


25.  9G  sq.  in. 

26.  108  sq.  in. 

27.  120  sq.  in. 


224 


DENOMINATE  AMOUNTS. 


476.  Oral  Exercise. 

A  square  one  yard  long  aiid  one  yard  wide  is  a  Square 
Yard  (sq.  yd.). 

1.  How  niuny  feet  long  is  a  square  one  yard  long  ? 

2.  IIow  many  feet  wide  is  a  square  one  yard  wide  ? 

This   drawing   represents   a 

square  yard  divided  into  square 
feet.  It  is  drawn  on  the  scale 
of  half  an  inch  to  the  foot. 

5.  What    does    each    small 
square  represent  ? 

4.  How    many   square    feet 
are  there  in  each  row  ? 

5.  How  manyrows  are  there? 

6.  How  many  square  feet  are  there  in  a  square  yard  ? 

7.  How  many  square  feet  are  there  in  J  of  a  square  yard  ? 

8.  How  many  square  feet  are  tliere  in  f  of  a  square  yard  ? 

9.  What  part  of  a  square  yard  is  1  sq.  ft.  ?     3  sq.  ft.  ? 
G  sq.  ft.? 

70.  How  many  square  feet  are  there  in  2  square  yards  ? 


477.  TABLE. 

144  square  inches  (sq.  in.)  =  1  square  foot  (sq.  ft.). 


9  square  feet 
30 J  square  yards 
IGO  square  rods 

10  square  cliains 
640  acres 


=  1  square  yard  (sq.  yd.). 
=  I  square  rod  (sq.  rd.). 

=  1  acre  (A.). 

=  1  squan^  mile  (sq.  mi.). 


Note.     A  square  rod  is  also  called  a  Perch. 


MEASURES  OF  SURFACE.  225 

478.  Exercise. 
/.  Draw  a  2-incli  square. 

Note.  By  a  2-inch  square  is  meant  a  square  2  in.  long  and 
2  in.  wide. 

2.  Divide  the  2-in.  square  into  square  inches.  How 
many  square  inches  are  there  in  1  row  ?  IIow  many  rows 
are  there  ?     How  many  square  inches  in  the  square  ? 

3.  Draw  a  3-in.  square  and  find  how  many  square  inches 
it  contains. 

4.  Draw  a  square  inch  and  divide  it  into  4  equal  squares. 

5.  How  long  is  each  part  ?     How  wide  ? 

6.  A  square  half  an  inch  long  and  half  an  inch  wide  is 
what  part  of  a  square  inch  ? 

7.  Draw  a  square  4  inclies  long  and  4  inches  wide. 

8.  How  many  square  inches  are  there  in  a  4-in.  sqnare  ? 

9.  How  many  square  feet  are  there  in  a  square  5  ft. 
long  and  5  feet  wide  ? 

10.  How  many  square  feet  are  there  in  a  square  6  ft. 
long  and  6  feet  wide? 

//.  How  many  square  yards  are  there  in  a  square  3  yd. 
long  and  2  yd.  wide  ? 

12.  Draw  a  square  5^  in.  long  and  5^  in.  wide.  Find 
the  number  of  sqnare  inches  in  tliis  sqnare. 

13.  If  each  inch  of  the  square  mentioned  in  problem  12 
represents  a  yard,  how  long  is  the  square  ?  How  wide  ? 
How  many  square  yards  in  it? 

14.  How  many  yards  long  is  a  square  rod  ?  How  many 
yards  wide  ? 

15 


226 


DENOMINATE  AMOUNTS. 


15.  How  many  square  yards  are  there  in  a  square  rod  ? 

16.  How  many  square  yards  are  there  in  4  sq.  rd.  ? 


479.  Exercise. 


20  rd. 


This  drawing  represents  a  square 
20  rd.  long  and  20  rd.  wide. 

/.  How  many  squares  each  1  rd.  long 
could  be  placed  along  one  side  of  this 
square  ? 

2.  How  many  rows  of  such  squares 
could  be  placed  in  the  whole  square  ? 

3.  How  many  such  squares  would  this  square  contain  ? 

4.  What  must  be  multiplied  together  to  give  the  number 
of  square  rods  in  this  square  ? 

5.  How  many  square  rods  are  there  in  an  acre  ? 

6.  What  part  of  an  acre  is  represented  by  this  square  ? 

7.  How  many  square  rods  are  there  in  a  square  6  rd. 
long  and  G  rd.  wide  ? 

8.  How  many  acres  are  there  in  each  of  these  squares  ? 


(c) 


(d) 


(b) 


(a) 


40  rd 


60  rd. 


80  rd.. 


90  rd. 


CUBIC  MEASURE. 


227 


Cubic  Measure. 
480.  Anything    that    has    length,    breadth,  and 
thickness  is  a  Solid. 


481.  A  Cube  is  a  solid  bounded 
by  six  equal  squares. 

482.  Oral  Exercise. 

/.  How  many  edges  of  this  fig- 
ure can  you  see  ?    How  many  faces  ? 


A  Cube. 


2.  How  many  edges  of  this  figure  can  you  not  see?     How 
many  faces  ? 

3.  How  many  edges  has  this  figure?     How  many  faces? 

4.  Measure  the  length  of  each  edge  that  you  can  see. 
Are  the  edges  equal? 

5.  What  is  the  name  of  this  figure? 

6.  When  a  cube  is  an  inch  long,  an  inch  wide,  and  an 
inch  high,  what  is  it  called  ? 

7.  If  each  edge  of  this  cube  were  a  foot  long,  what  would 
the  cube  be  called? 

8.  If  each  edge  of  this  cube  were  a  yard  long,  what 
would  the  cube  be  called? 

483.  Oral  Exercise. 

This  figure  represents  a  cubic  foot 
with  one  layer  of  cubic  inches  in  it. 

/.  How  many  cubic  inches  are  there 
in  1  row  of  this  layer? 

2.  How  many  rows  of  cubic  inches 
are  there  in  this  layer? 


228 


DENOMINATE  AMOUNTS. 


3.  How  many  cubic  inches  are  there  in  this  layer? 

4.  ITow  many  iiiclies  high  is  a  cubic  foot? 

5.  How  many  layers  of  cubic  inches  can  be  placed  in  a 
cubic  foot? 

6.  How  many  cubic  inches  are  there  in  a  cubic  foot? 

This  figure  represents  a  cubic 

yard  divided  into  cubic  feet. 

7.  How  many  cubic  feet  are 
there  in  one  row  of  the  upper 
layer? 

8.  How  many  rows  of  cubic 
feet  are  there  in  the  upper 
layer? 

9.  How  many  cubic  feet  are  there  in  the  upper  layer? 

10.  How  many  layers   of  cubic   feet  are  there  in  the 
cube? 

//.  How  many  cubic  feet  are  there  in  a  cubic  yard? 


484. 


TABLE. 


1728  cubic  inches  (cu.  in.)  =  1  cubic  foot  (cu.  ft.). 
27  cubic  feet  =  1  cubic  yard  (cu.  yd.). 


485. 


Important  Facts. 


1  gallon  =  231  cu.  in. 
1  bushel  =  2150.42  cu.  in. 
1  bushel  =  '^i  cu.  ft.  nearly. 
62J  lb.  =  the  weight  of  1  cu.  ft.  of  water. 
24 J  cu.  ft.  =  1  perch  of  masonry. 


CUBIC  MEASURE.  229 

486.  Oral  Exercise. 

/.  How  many  cubic  feet  are  there  in  2  cu.  yd.  ? 

2.  How  many  cubic  feet  are  there  in  |  cu.  yd.? 

3.  What  part  of  a  cubic  yard  is  9  cubic  feet? 

4.  What  part  of  a  cubic  yard  is  18  cubic  feet? 

5.  How  many  cubic  feet  are  there  in  a  bin  that  will  hold 
4bu.?     [§485.] 

6.  How  many  cubic  feet  are  there  in  a  bin  that  will  hold 
10  bu.? 

7.  A  bin  containing   20  cu.  ft.  will   hold   how  many 
bushels? 

487.  Exercise. 


This  drawing  represents  a  cube  5  in. 
long,  5  in.  wide,  and  5  in.  high. 

/.  How  many  cubic  inches  could  be 
placed  in  this  cube  along  one  edge?  How 
many  rows  would  make  one  layer  of  cubic  inches  in  the 
cube?  How  many  such  layers  could  be  placed  in  the  cube? 
How  many  cubic  inches  would  this  cube  make?  What  did 
you  multiply  together  to  get  the  number  of  cubic  inches 
the  cube  would  make? 

2.  How  many  cubic  inches  are  there  in  a  cube  6  in.  long? 

3.  Draw  a  cube  2   in.  long  and   find   the  number  of 
cubic  inches  in  it. 

4.  How  many  cubic  feet  are  there  in  a  cube  2  ft.  long? 

5.  How  many  cubic  yards  are  there  in  a  cube  4  yd. 
long  ? 


^^\ 

^ 

1 
j 

1 
i 

J 

y^ 

230 


DENOMINATE  AMOUNTS. 


6.  Find  the  number  of  cubic  feet  in  each  of  the  cubes 
represented  by  these  drawings: 

(c) 


(a) 


o 


y^\ 

"7 

(b) 

! 

5 

1 

i 
1 

} -- 

/^i 

y 

i 
1 

/ 

/ 

3  ft. 


5  ft. 


8  ft. 


Rectangle 


Rectangles. 

488.  A  Rectangle  is  a  surface  that 
has  four  sides  and  four  right  an- 
gles. 

The  opposite  sides  of  a  rectangle 
are  equal. 


489.  Exercise. 

ABCD  represents  a  rectangle  4  in. 
long  and  3  in.  wide,  divided  into 
square  inches. 

/.  How  many  square  inches  are 
there  in  one  row  of  ABCD  ?  How  many  rows  are  there 
in  ABCD  ?     How  many  square  inches  ? 

2.  What  numbers  did  you  multiply  together  to  get  the 
number  of  square  inches  in  rectangle  ABCD? 

3.  If  each  square   in   ABCD  were  a  square  foot,  how 


RECTANGLES. 


231 


long  would  ABGD  be?    How  wide?    How  many  square 
feet  would  it  contain? 

4.  Draw  a  rectangle  5  in.  long  and  2  in.  wide,  divide  it 
into  square  inches,  and  find  how  many  square  inches  there 
are  in  it. 

5.  Draw  a  rectangle  4  in.  long  and  ^  in.  wide,  and  find 
how  many  square  inches  it  contains. 

6.  Find  how  many  square  feet  there  are  in  each  of  the 
rectangles  represented  here : 


8  ft.  10  ft.  12  ft. 

490.  Find  the  number  of  square  inches  in  a  window 
pane  10  in.  long,  and  8  in.  wide. 

10  =  the  number  of  inches  in  the  length  of  the  pane. 
8=   "         "        "       "       "     "    width    ''    ''      " 
.'.  8x  10,  or  80  =  "         "        "  square  inches  in  the  pane. 
Note.     The  sign  .  •.  is  read  therefore. 

491.  Exercise. 

/.  How  many  square  feet  of  boards  are  there  in  a  floor 
12  ft.  long  and  9  ft.  wide  ? 

2.  How  many  square  yards  of  bunting  will  it  take  to 
make  a  flag  3J  yd.  long  and  If  yd.  wide  ? 

5.  How  many  square  rods  are  there  in  a  lot  60^  rd.  long 
and  14  rd.  wide  ? 

4.  How  many  square  miles  are  there  in  a  rectangular 
tract  of  land  5J  mi.  long  and  3J  mi.  wide  ? 


232 


DENOMINATE  AMOUNTS. 


6.  How  many  square  yards  of  oil  cloth  will  it  take  to 
cover  a  floor  15  ft.  long  and  12  ft.  wide  ? 

6.  How  many  acres  are  there  in  a  rectangular  field  80 
rd.  long  and  20  rd.  wide  ? 

7.  How  many  square  yards  of  plastering  are  there  in  the 
end  wall  of  a  room  12  ft.  wide  and  9  ft.  high,  if  it  contains 
one  door  7^  ft.  long  and  4  ft.  wide  ? 

8.  How  many  shingles  are  required  to  cover  a  roof  20  ft. 
long  and  16  ft.  wide,  if  each  shingle  covers  20  sq.  in.  of 
roof  ? 


15  a 


Carpeting. 

492.  In  determining  the  num- 
ber of  yards  of  carpet  that  must 
be  bought  to  carpet  a  floor, 
the  number  of  strips  needed 
should  first  be  found  ;  a  frac- 
tional part  of  a  strip  should  be 
regarded  as  a  full  strip. 

493.  How  many  yards  of  car- 
pet }  yd.  wide  must  be  bought  to 
carpet  a  floor  24  ft.  long  and  15 
ft.  wide,  the  strips  to  be  laid 
lengthwise  ? 


15  ft.,  or  180  in.  =  the  width  of  the  floor. 

i  yd.,  or  37  in.  =  tlie  width  of  a  strip  of  carpet. 

180  +  27,  or  05  =  the  number  of  strips  needed. 

Then,  7  strips  must  be  bought. 

24  ft.,  or  8  yd.  =  the  length  of  a  strip. 

Then,  7x8,  or  56  =  the  number  of  yards  required. 


LUMBER  MEASURE.  233 

494.  Exercise. 

/.  How  many  strips  of  carpet  1  yd.  wide  must  be  bought 
for  a  room  12  ft.  wide,  the  strips  to  be  laid  lengthwise  ? 

2.  How  many  strips  of  carpet  J  yd.  wide  must  be 
bought  for  a  room  27  ft.  long,  the  strips  to  be  laid  cross- 
wise ? 

3.  How  many  strips  of  carpet  J  yd.  wide  must  be  bought 
for  a  room  20  ft.  long,  18  ft.  wide  : 

1st,  if  the  strips  are  to  be  laid  lengthwise  ? 
2d,  if  the  strips  are  to  be  laid  crosswise  ? 

4.  How  many  yards  of  carpet,  1  yd.  wide,  are  needed  to 
cover  a  floor  12  ft.  long  and  9  ft.  wide  ? 

5.  How  many  yards  of  carpet,  J  yd.  wide,  are  needed  to 
cover  a  floor  18  ft.  long  and  13J  ft.  wide  ? 

6.  Find  the  cost  of  carpeting  a  floor  24  ft.  long  and  18 
ft.  wide  with  carpet  |  yd.  wide,  costing  10.90  per  yard, 
strips  to  be  laid  lengthwise. 

7.  Find  the  number  of  yards  of  carpet  |  yd.  wide  that 
must  be  bought  to  carpet  40  rooms  in  a  normal  school, 
each  16  ft.  long  and  llj  ft.  wide,  strips  to  be  laid 
lengthwise. 

Lumber  Measure. 

495.  Lumber,  one  or  more  inches  thick,  is  meas- 
ured by  the  board  foot. 

496.  A  Board  Foot  is  a  piece  of  lumber  one  foot 
long,  one  foot  wide,  and  one  inch  thick. 

Note.  A  board  one  inch  thick  contains  as  many  board  feet  as  it 
does  square  feet. 


234  DENOMINATE  AMOUNTS. 

497.  Find  the  number  of  board  feet  in  a  board  16 
feet  long  and  5  in.  wide. 

16  =  the  number  of  feet  in  the  length. 

A=    "         "        "    "     "    "   width. 

•  *.  'h  X  16,  or  '-^  =  the  number  of  board  feet  in  the  board. 

Therefore,  to  fintl  the  number  of  board  feet  in  a 
board  one  inch  thick  : 

Multiply  the  number  of  feet  in  the  length  by  the 
number  of  inches  in  the  width  and  divide  by  12, 

498.  Find  the  number  of  board  feet  in  a  plank  15 
feet  long,  7  inches  wide,  and  2  inches  thick. 

'^■~   =  the  number  of  board  feet  in  the  plank,  if  1  inch  thick. 
.  2x7x15^    u        u     u         u     u    .<       .«        u       2  inches  thick. 

•    •  12 

Therefore,  to  find  the  number  of  board  feet  in  a 
piece  of  lumber  more  than  one  inch  thick  : 

Multiply  together  the  number  of  feet  in  the 
lengthy  the  number  of  inches  in  the  widths  and  the 
number  of  inches  in  the  thickness,  and  divide  by  P2, 

Note  1.     Boards  less  than  one  inch  thick  are  sold  by  the  square  foot. 
Note  3.     The  number  of  board  feet  in  a  piece  of  lumber  is  spoken 
of  as  the  number  of  feet. 

499.  Exercise. 

/.  Find  the  number  of  feet  in  a  board  16  ft.  by  8  in. 

2.  Find  the  number  of  feet  in  a  plank  15  ft.  long,  9  in. 
wide,  and  2  in.  thick. 

3.  Find  the  number  of  feet  in  a  piece  of  scantling  14  ft. 
long,  4  in.  wide,  and  3  in.  thick. 


RECTANGULAR  SOLIDS. 


235 


4.  How  many  square  feet  are  there  in  a  board  12  ft. 
long,  6  in.  wide,  and  ^  of  an  inch  thick  ? 

6.  How  many  feet  in  the  following  bill  of  lumber  ? 

8  boards,  18  ft.  by  8  in. 

10  boards,  16  ft.  by  8  in. 

12  boards,  14  ft.  by  9  in. 

6.  How  many  feet  in  a  load  of  lumber  containing  the 
following  ? 

3  joists,  16  ft.  by  9  in.  by  4  in. 

4  scantlings,  14  ft.  by  4  in.  by  3  in. 
2  planks,  12  ft.  by  8  in.  by  2J  in. 

8  boards,  18  ft.  by  9  in. 

7.  Find  the  cost  of  : 

6  boards,  16  ft.  by  8  in.,  at  $30  per  M  (1000  ft.). 
8  pieces  of  scantling,  18  ft.  by  4  in.  by  3  in. ,  at  $25  per  M. 
'  4  planks,  12  ft.  by  9  in.  by  2 J  in.,  at  $28  per  M. 


Rectangular  Solids. 

600.  A  Rectangular  Solid  is  a 
solid  all  of  whose  faces  are 
rectangles. 


A  Rectangular  Solid. 


501.  Exercise. 

/.  How  many  faces  has  solid 

\(    ?     How  many  of  them  can 

}   u  see  ?     What  kind  of  figure 

1      lach  face  ?     What  may  the 

solid  be  called  ? 

2.  AC  is  divided  into  equal 


236 


DENOMINATE  AMOUNTS. 


cubes.  How  many  ©f  these  cubes  are  there  in  each  row 
of  the  upper  layer  ?  How  many  rows  are  there  in  the 
upper  layer  ?  How  many  cubes  are  there  in  the  upper 
layer  ?     How  many  layers  of  cubes  are  there  in  AC  ? 

3.  If  each  cube  of  AC  were  a  cubic  incli,  how  long 
would  the  solid  AC  be  ?  How  wide  ?  How  high  ?  How 
many  cubic  inches  would  it  contain  ? 

4.  What  three  numbers  did  you  multiply  together  to 
get  the  number  of  cubes  in  AC  ? 

5.  If  the  rectangular  solid  were  4  feet  long,  2  feet  wide, 
and  3  feet  high,  show  that  the  number  of  cubic  feet  in  it 
would  be  3  X  2  X.  4. 


502.  A  pile  of  wood 
or  stone  8  ft.  long,  4  ft. 
wide,  and  4  ft.  high  is 
called  a  Cord. 


603.  Exercise. 

/.  How  many  cubic  inches  are  there  in  a  block  5  in. 
long,  4  in.  wide,  and  3  in.  high  ? 

5  =  the  number  of  inches  in  the  lengtli  of  tlie  block. 

4  _.     u         ((         u        u       u     <4    width    "    "         ** 

3  _      w  a  a         a         u      u     ]]^,\crhl    a     «'  « 

,-.  3  X  4  X  5,  or  GO  —  the  number  of  cubic  inches  in  the  block. 

2.  How  many  oul)io  feot  of  air  are  there  in  a  room  12  ft. 
long,  9  ft.  wide,  and  8  ft.  Iiigh  ? 

3.  How  many  cubic  yards  of  earth  must  be  removed  in 
^liggi^ig  a  cellar  27  ft.  long,  18  ft.  wide,  and  12  ft.  deep  ? 


COMPOUND  DENOMINATE  AMOUNTS.  237 

4.  Show  that  there  are  128  cu.  ft.  in  a  cord.     [See  §502.] 

5.  Show  that  a  wall  16^  ft.  long,  IJ  ft.  thick,  and  1  ft. 
high  contains  24|  cu.  ft.,  or  1  perch.     [See  §485.] 

6.  Show  that  a  tin  vessel  11  in.  long,  7  in.  wide,  and 
3  in.  deep,  holds  a  gallon.     [See  §485.] 

7.  How  many  gallons  of  water  will  a  cistern  hold  that  is 
28  ft.  long,  22  ft.  wide,  and  12  ft.  deep  ? 

8.  If  1  bushel  equals  about  IJ  cu.  ft.,  about  how  many 
bushels  will  a  granary  hold  that  is  10  ft.  long,  8  ft.  wide, 
and  6  ft  deep  ? 

9.  If  a  heaped  bushel  equals  f|  cu.  ft.,  how  many  heaped 
bushels  will  a  corncrib  hold  that  is  25  ft.  long,  4  ft.  wide, 
and  10  ft.  high  ? 

10.  Potatoes  are  sold  by  the  heaped  bushel ;  about  how 
many  bushels  of  potatoes  will  a  bin  hold  that  is  10  ft.  long, 
8  ft.  wide,  and  5  ft.  deep  ? 

//.  How  many  cords  of  wood  are  there  in  a  pile  256  ft. 
long,  4  ft.  wide,  and   4  ft.  high  ? 

Compound    Denominate   Amounts. 

504.  Find  the  sum  of  4  bu.  3  pk.  2  qt.,  3  bu.  1  pk. 
5  qt.,  and  2  bu.  2  pk.  6  qt. 

The   sum  of  the  quarts  is  13  qt.,  or  1  pk. 

bu.     pk.     qt.  5  qt.     Write  5  under  the  quarts  and  add  1  pk. 

4       3         2  to  the  pecks.     The  sum  of  the  pecks  is  7  pk., 

'^        1         ^  or  1  bu.  3  pk.      Write  3  under  the  pecks  and 

'^       ^         ^  add  1  bu.   to  the  bushels.     The  sum  of  the 


10       3        5  bushels  is  10  bu. 

.*.  The  sum  is  10  bu.  3  pk.  5  qt. 


238  DENOMINATE  AMOUNTS. 


yr. 

mo. 

da. 

20 

6 

18 

14 

8 

14 

605.  From  20  yr.  G  mo.  18  da.  take  14  yr.  8  mo.  14  da. 
14   da.    from   18  da.  leaves  4  da.     Write  4 

under  the  days.  We  cannot  take  8  mo.  from 
6  mo.,  so  we  take  1  yr.,  or  12  mo.,  from  20  yr. 
(leaving  19  yr.)  and  add  it  to  6  mo.,  making 
18  mo.  8  mo.  from  IS  mo.  leaves  10  mo.  Write 
5  10  4  10  under  the  months.  14  yr.  from  19  yr.  leaves 
5  yr. 

.-.  The  difference  is  5  yr.  10  mo.  4  da. 

606.  A  man  who  was  born  Fob.  10,  1872  and  died  Aug. 
17,  1903,  lived  to  what  age  ? 

He  was  born   on   the   10th    day  of  the  2d 

yr.     mo.  da.  month  of  the  year  1872,  and  died  on  tlie  17th 

1903    8       17  (Jay  of  the  8th  mo.  of  the  year  1903. 

•^^^^    ^       ^^  .-.  1903  yr.  8   mo.  17  da. -1872  yr.    2  mo. 

31    0         7  10  da.,  or  31  yr.  6  mo.  7  da.,  was  the  age  to 

which  he  lived. 


)7.  Exercise. 

Add: 

1. 

Add 

: 

5. 

ft. 

in. 

T. 

cwt. 

lb. 

3 

8 

2 

13 

48 

6 

9 

1 

7 

82 

9 

10 

2 

8 

10 

Add: 

2. 

4. 

lb. 

oz. 

pwt. 

gal. 

qt. 

pt 

3 

!) 

5 

From 

10 

2 

0 

4 

8 

12 

take 

G 

3 

1 

5 

10 

1() 

COMPOUND  DENOMINATE  AMOUNTS.  239 


5. 

6. 

cu.  yd. 

cu.  ft. 

yd. 

ft. 

in. 

From  25 

10 

From  16 

2 

3 

take  14 

24 

take    8 

2 

9 

7.  In  one  bag  there  are  2  bu.  3  pk.  of  potatoes,  and  in 
another  1  bu.  2  pk.;  what  is  the  amount  of  potatoes  in 
the  two  bags? 

8.  A  keg  contained  5  gal.  2  qt.  of  cider,  but  2  gal.  3 
qt.  has  been  drawn  off;  how  much  remains? 

9.  A  man  was  born  Jan.  12,  1882,  and  had  his  life  in- 
sured on  Feb.  9,  1902;  how  old  was  he  when  his  life  was 
insured  ? 

10.  George  Washington  was  born  Feb.  22,  1732,  and 
died  Dec.  14,  1799;  how  old  was  he  when  he  died? 

//.  Daniel  Webster  was  born  Jan.  18,  1782,  and  died 
Oct.  24,  1852;  how  old  was  he  when  he  died? 

12.  William  Penn  was  born  Oct.  14,  1644,  and  died  July 
30,  1718;  how  old  was  he  when  he  died? 

13.  Henry  Clay  died  June  29,  1852,  at  the  age  of  75  yr. 
2  mo.  17  da.;  when  was  he  born? 

508.  Reduce  3  gal.  2  qt.  1  pt.  to  pints. 

3  -  2  -•  1. 
_4  1  gal.  =  4  qt. 

■^n  3  gal.  =  3  X  4  qt.,  or  12  qt. 

14  3  gal.  3  qt.  =  12  qt.  +  2  qt.,  or  14  qt. 

^  1  qt.  =  3  pt. 

28 
2  14  qt.  =  14  X  3  pt.,  or  38  pt. 

29  •••  3  gal.  2  qt.  1  pt.  =  38  pt.  +  1  pt.,  or  39  pt. 


240  DENOMINATE  AMOUNTS. 

509.  Exercise. 

/.  Reduce  7  lb.  2  oz.  (avoir.)  to  ounces. 

2.  Ueduce  4  T.  8  cwt.  to  hundredweight. 

3.  Reduce  4  du.  8  hr.  to  hours. 

4.  Reduce  4  A.  30  sq.  rd.  to  square  rods. 

5.  Reduce  12  pwt.  10  gr.  to  grains. 

6.  Reduce  2  mi.  18  rd.  to  rods. 

7.  Reduce  4  bu.  2  pk.  G  qt.  to  quarts. 

8.  Reduce  2  yd.  2  ft.  6  in.  to  inches. 

610.  Reduce  149  pt.  to  gallons,  etc. 

2  )  149  i  of  the  number  of  pt.  =  the  number  of  (jt. 

4)74-^  .-.  149  pt.  =  74  qt.  1  pt. 

18^  i  of  the  number  of  qt.  =r  tlie  number  of  gal. 

.-.  74  qt.  =  18  gal.  2  qt. 
Hence,  149  pt.  —  18  gal.  2  qt.  1  pt. 

611.  Exercise. 

/.  Reduce  3675  sec.  to  hours,  etc. 

2.  Reduce  500  gi.  to  gallons,  etc. 

5.  Reduce  150  qt.  to  bushels,  etc. 

4.  Reduce  4875  lb.  to  tons,  etc. 

5.  Reduce  275  pt.  to  gallons,  etc. 

6.  Reduce  2000  yd.  to  miles,  etc. 


CHAPTER  y. 
PERCENTAGE. 

512.  Instead  of  saying  five  Jiundredths  of  a  mim- 
ber,  we  may  ^^ij  five  per  cent  of  a  number. 

Per  cent  means  hundredths. 

Thus,       1  per  cent  of  a  number  is  j^tr,  of  it. 

4  per  cent  of  a  number  is  i  Jo,  or  -gV  of  it. 
25  per  cent  of  a  number  is  -j^,^)-,  or  \,  of  it. 
100  per  cent  of  a  number  is  \%%  or  once,  tlie  number. 
200  per  cent  of  a  number  is  fHo,  or  twice,  the  number. 
250  per  cent  of  a  number  is  fooi  or  2^  times,  tlie  nmiiber. 
12]  per  cent  of  a  number  is  -J  J*,  or  ^,  of  it. 

513.  The  sign  i  stands  tor  per  cent. 

Thus,  5  per'  cent  of  60  may  be  expressed  5  %  of  50. 

514.  Oral  Exercise. 

Wbat  part  of  a  number  is  : 


/.     2^  of  it?      5.  15  fo  of  it? 

5. 

30  ^  of  it  ? 

2.     4^  of  it?       6.  40 

^  of  it  ? 

10. 

80  ^  of  it  ? 

3.   10  fc  of  it  ?       7.   50  ^  of  it  ? 

11. 

00  ^  of  it  ? 

4.  20  ^  of  it  ?       8.  75 

fo  of  it  ? 

12. 

45  ^  of  it  ? 

515.  Oral  Exercise. 

Express  as  a  per  cent : 

/.   }.          3.   i.          5.   |. 

7-  A. 

9. 

|.          //.  h 

2.   J.          4.   |.          6.   A 

«•  tV 

10. 

IJ.        12.  i. 

616.  Find  5  ^  of  40. 

5^  of  40  =  j^,  or  -A-,  of  40, 

or  2. 

10 

242 


PERCENTAGE. 


517.  Oral  Exercise. 

Find: 
/.     1  ^  of  200. 

2.  50  i  of  12. 

3.  25  i  of  8. 

4.  20  i  of  35. 

5.  10  i  of  70. 
5.  5  ^  of  80. 
7.     2  ^  of  150. 


8.  4:fc  of  100. 

5.  40  fc  of  15. 

/O.  00  ^  of  50. 

//.  30^  of  30. 

12.  90  ^  of  90. 

13.  75  ^  of  24. 
/4.  8  ^  of  50. 


/5.  25  ^  of  $10. 

/^.  12J  ^  of  $0.40. 
/7.       4  ^  of  $200. 

18.  2^  of  $1000. 

19.  100  ^  of  $2. 

20.  200^  of  $1.50. 

21.  300  ^  of  $3. 


518.  2  is  25^  of  what  number  ? 

2  =  i^tfo,  or  ^,  of  the  required  number. 

.'.  f  of  the  required  number  =  4  x  2,  or  8. 

519.  Oral  Exercise. 

Find  the  number  of  which  : 


/.  0  is  50^.  6.  14  is  200^. 

2.  8  is  25^.  7.  4  is  80j^. 

3.  3  is  20^.  8.  20  is  400^^. 

4.  2J  is  10^.  9.  18  is  00^. 

5.  5  is    1^.  10.  30  is  90^. 

520.  What  per  cent  of  20  is  8  ? 

8  =  A,  or  I  of  20. 

.  •.  8  =  ^  of  100  ^  of  20,  or  40  %  of  20. 

521.  Oral  Exercise. 

What  per  cent  of  : 


//.  10  is  12J^. 

12.  15  is  75^. 

13.  1.5  is  50^. 

14.  I  is  (>0^. 

15.  8  is  33J^. 


/.  4  is    2? 

2.  5  is    3  ? 

3.  10  is    1  ? 

4.  75  is  25  ? 

5.  50  is    2? 


6.  100  is  5  ? 

7.  30  is  0  ? 
5.  200  is  4? 
9.  2  is  4  ? 

10.  9  is  9  ? 


//.  50  is  40? 

12.  80  is  GO  ? 

13.  1  is    J  ? 

14.  5  is  2J  ? 
/5.  i  is    J? 


PERCENTAGE.  243 


622.  Oral  Exercise. 

/.  How  much  is  20^  of  120  ? 

2.  How  much  is  H  of  50  ct.  ? 

3.  A  farmer  sold  50^  of  80  bu.  of  wheat ;  how  much 
wheat  did  he  sell  ? 

4.  $18  is  f  of  what  sum  ?     75^  ©f  what  sum  ? 

5.  $20  is  I  of  what  sum  ?     80^  of  what  sum  ? 

6.  2  yd.  is  50 fo  of  what  length  ? 

7.  2  lb.  is  bfo  of  what  weight  ? 

8.  50  ct.  is  2  times  what  sum  ?    200^  of  what  sum  ? 

9.  6  ft.  is  3  times  what  length  ?     300^  of  what  length  ? 
10.  25  lb.  is  200^  of  what  weight  ? 

11.1  qt.  is  what  part  of  4  qt.  ?     AVhat  per  cent  of  4  qt.? 

12.  $2  is  what  part  of  $4  ?     What  per  cent  of  14  ? 

13.  8  ct.  is  what  part  of  20  ct.  ?  What  per  cent  of 
20  ct.  ? 

14.  1  pk.  is  what  part  of  1  bu.?  What  per  cent  of 
1  bu.? 

15.  10  ct.  is  what  part  of  $1  ?  What  per  cent  of 
of  $1? 

16.  G  ct.  is  what  part  of  $1  ?    What  per  cent  of  II  ? 

623.  Oral  Exercise. 

/.  A  boy  was  given  50  words  to  spell ;  if  he  spelled  80^ 
of  them  correctly,  how  many  did  he  spell  correctly  ? 

2.  75j^  of  a  class  of  40  pupils  were  promoted  ;  how 
many  were  promoted  ? 

5.  A  dealer  bought  a  horse  for  $200  and  sold  him  so  as 
to  gain  20^  of  tlie  cost ;  what  did  he  receive  for  him  ? 


344  PERCENTAGE. 

4.  A  farmer's  corn  crop  was  160  bu.;  he  used  37J,<^  of  it 
and  sold  tlie  remainder  ;  how  many  bushels  did  he  sell  ? 

6.  A  piece  of  silverware  weighing  8  oz.  is  02^^  pure 
silver  ;  how  many  ounces  of  silver  does  it  contain  ? 

6.  A  merchant  asked  $G  for  a  coat,  but  sold  it  at  20j^ 
below  this  price  ;  what  ^lid  he  receive  for  it  ? 

7.  A  boy  spelled  correctly  20  words,  which  were  80^ 
of  the  number  given  him  ;  how  many  were  given  him  ? 

8.  A  man  earned  $40  collecting  bills  that  were  overdue  ; 
how  much  did  he  collect,  if  he  was  paid  25^^  of  the  sum 
collected  ? 

9.  After  a  man  sold  60^  of  his  land,  he  had  left  20  A.; 
how  much  had  he  at  first  ? 

10.  A  laborer's  wages  were  increased  from  $10  to  $12  a 
week ;  find  the  per  cent  of  increase. 

//.  What  per  cent  should  a  girl  receive  who  spells 
correctly  20  words  of  the  25  given  ? 

12.  A  boy  attended  school  15  days  out  of  20  during  a 
certain  month  ;  find  the  per  cent  of  his  attendance. 

13.  A  merchant  who  buys  oil  at  8  ct.  a  gallon  and  sells 
it  at  10  ct.  a  gallon,  gains  what  per  cent  of  the  cost  ? 

524.  A  dealer  bought  a  horse  for  $175  and  sold  him  at 
a  gain  of  15^  of  the  cost ;  find  the  gain. 

$175 

ll^  $175  =  the  cost. 

875  15jf,  or  .15,  of  tlie  cost  =  the  gain. 

175  .*.  .15  X  $176,  or  $26.25  =  the  gain. 


$26.25 


PERCENTAGE.  245 


525.  Exercise. 

/.  A  man  had  $250  in  bank  and  withdrew  20^  of  it; 
how  much  did  he  withdraw  ? 

2.  A  dealer  bought  a  cow  for  $42  and  sold  her  at  a  gain 
of  37J^  of  the  cost ;  find  the  selling  price. 

Sdg.  37|^  of  a  number  =  .375  of  it. 

3.  Milk  yields  butter  to  the  extent  of  about  4;^  of  its 
weight ;  how  much  butter  will  175  lb.  of  milk  yield  ? 

4.  10^  of  all  silver  coin  is  copper ;  find  the  weight  of 
the  copper  in  a  silver  dollar,  which  weighs  412.5  gr. 

5.  A  merchant  agreed  to  settle  some  outstanding  bills 
amounting  to  1385.25  for  75^  of  their  full  amount;  how 
much  did  he  lose  ? 

6.  A  merchant  asked  |>G.25  for  a  coat,  but  sold  it  for  00^ 
of  the  price  asked  ;  how  much  did  he  receive  for  it  ? 

7.  A  dealer  bought  a  china  dinner  set  for  137.50  and 
asked  for  it  20^  more  than  the  cost ;  find  the  price  asked. 

8.  A  farmer  had  672  bu.  of  oats,  but  sold  16|^  of  it  and 
fed  25^  of  the  remainder  ;  how  many  bushels  had  he  left  ? 

526.  A  dealer  sold  a  horse  at  a  gain  of  130.00,  which 
was  15^  of  its  cost  ;  find  the  cost. 

.15)  $30.00  ($304  $30.60  =  the  gain. 

30  15^,  or  .15,  X  cost  =  the  gain. 
60  .-.  .15  X  cost  =  $30.60. 

60  Cost  =  $30.60-^.15,  or  $204. 

527.  Exercise. 

/.  A  merchant  failing  in  business  paid  75^  of  his  debts; 
if  he  paid  $3757.80,  what  were  his  debts? 


246  PERCENTAGE. 


2.  A  dealer  sold  a  machine  for  $126,  which  was  90ji^  of 
the  cost ;  find  the  cost. 

5.  A  jeweler  sold  a  clock  for  $15.50,  which  was  80j^  of 
the  price  he  asked  for  it  ;  what  did  he  ask  for  it  ? 

4.  An  agent  earned  $275.50  by  collecting  rents,  being 
paid  2^  of  the  sum  collected  ;  how  much  did  he  collect  ? 

5.  A  suit  of  clothes  was  sold  for  $2.25  more  than  it  cost, 
which  was  at  a  gain  of  12^  of  the  cost ;  find  the  cost. 

6.  A  merchant  sold  a  bureau  for  $24,  which  was  80ji^  of 
the  price  he  asked  ;  what  was  the  price  asked  ? 

7.  If  milk  yields  4^  of  its  weight  of  butter,  how  many 
pounds  of  milk  will  yield  150  lb.  of  butter  ? 

8.  A  farmer  sold  437.5  bu.  of  wheat,  which  was  87J^  of 
his  wheat  crop  ;  find  his  wheat  crop. 

Sua.  87^^  of  his  crop  =  .875  x  his  crop. 

628.  A  clerk  who  earned  $640  a  year,  paid  $96  a  year 
for  house  rent;  what  per  cent  of  his  salary  did  he  pay 
for  house  rent  ? 

6.40)96.0(15  1640  =  his  salary. 

^^  .01  of  $640,  or  $6.40  =  the  rent,  if  it  were  \%  of  the 

320  salary. 

320         •*•  $96:  $6. 40,  or  15= the  number  of  per  cent  required. 

629.  Exercise. 

/.  If  in  300  lb.  of  corn  there  are  16.2  lb.  of  fat,  what  is 
the  per  cent  of  fat  in  corn  ? 

2.  A  music  dealer  bought  a  second-hand  piano  for  $275 
and  sold  it  at  a  gain  of  $49.50  ;  what  per  cent  of  the  cost 
did  he  gain  ? 


INTEREST.  247 


5.  A  marketman  sold  for  a  merchant  178  worth  of 
poultry,  charging  $3.90  for  making  the  sale;  what  per 
cent  of  the  selling  price  was  charged  ?  ' 

4.  A  lawyer  collected  a  debt  of  $1200,  charging  $150 
for  making  the  collection  ;  what  per  cent  of  the  debt  did 
he  charge  ? 

5.  I  bought  a  bill  of  goods  amounting  to  $112.50,  but 
the  merchant  deducted  14.50  from  it  for  cash  payment ; 
what  per  cent  of  the  bill  did  he  deduct  ? 

6.  A  shoe  dealer  paid  $3  for  a  pair  of  gum  boots,  and  sold 
them  for  $2.40  ;  what  per  cent  of  the  cost  did  he  lose  ? 

7.  A  hardware  dealer  bought  nails  at  $2.40  a  keg  and 
sold  them  at  $2.80  a  keg;  what  per  cent  of  the  cost  did 
he  gain  ? 

8.  A  butcher  bought  pork  at  $5.40  per  cwt.  and  sold  it 
at  $6.30  per  cwt.;  what  per  cent  of  the  cost  did  he  gain  ? 


Interest. 

630.  If  a  person  borrows  money  he  usually  has  to 
pay  a  certain  per  cent  of  the  sum  borrowed  for  the 
use  of  it  each  year. 

Money  paid  for  the  use  of  money  is  called  Interest. 

The  sum  on  which  interest  is  paid  is  called  the 
Principal. 

Thus,  if  a  man  borrows  $100  for  3  years  and  has  to  pay  5%  of 
tliis  sum  for  the  use  of  it  each  year,  lie  must  pay  2  times  5%  of  $100, 
or  $10,  for  the  use  of  it  for  2  years.  Here  $10  is  the  interest  and 
$100  the  principal. 


S48  PERCENTAGE. 

631.  Oral  Exercise. 

What  is  the  interest  on: 

/.  $100  for  1    yr.  at  6^?  6.  $500  for  ^  yr.  at  6^? 

2.  $200  for  1    yr.  at  5^?  7.  $250  for    ^  yr.  at  4^? 

5.  $600  for  2    yr.  at  4^?  8.  $800  for  1^  yr.  at  4^? 

4.  $300  for  3    yr.  at  G^?  5.  $000  for  2|  yr.  at  3^? 

5.  $100  for  2^  yr.  at  4^?  /O.  $000  for  1  mo.  at  0^? 

532.  The  time  for  which  interest  is  paid  is  usually 
reckoned  in  years  and  days,  when  it  exceeds  a  year; 
and  in  days  only  when  it  is  less  than  a  year^  the 
exact  number  of  days  being  counted. 

Thus,  if  a  sum  of  mouey  is  on  interest  from  Jan.  2,  1901  to 
March  1,  1903,  the  time  is  counted  as  follows:  from  Jan.  2,  1901 
to  Jan.  2,  1903  is  2  yr. ;  and  from  Jan.  2,  1903  to  Mar.  1,  1903  is 
58  da.,  making  in  all  2  yr.  58  da.  Again,  if  a  sum  of  money  is  on 
interest  from  Mar.  1,  1903  to  Oct.  1,  1903,  the  time  is  214  da. 

533.  Interest  for  time  less  than  a  year  is  usually 
reckoned  on  the  basis  of  360  da.  to  the  year. 

Thus,  in  reckoning  interest  on  a  sum  for  108  da.,  the  time  is 
usually  regarded  as  ^gg  yr. 

534.  Find  the  interest  on  $175  from  May  1,  1901  to 
Sept.  1,  1903,  at  4^. 

$175  The  time  from  May  1,  1901  to  May  1, 1903  =  2  yr. 

'0^  The  time  from  May  1, 1903  to  Sept.  1,  1903  = 

^^'^^  (30  +  30  +  31  +  31  +  1)  da.  =  123  da. 

2  Vn- 

^^^    .'.  The  time  the  principal  is  on  interest  = 

2.;j9j^  2  yr.  123  da.  =  2iV5  yr. 

$lG.39i  -^^  ^  $175,  or  $7  =  tlie  interest  for  1  yr. 

2,V..  > 
interest. 


BUSINESS  PAPERS. 


249 


635.  Exercise. 

Find  the  interest  on: 

/.  $50  for  2  yr.  at  6^. 

2.  $75  for  3  yr.  at  4^. 

3.  $150  for  2  yr.  at  ^fo. 

4.  $350  for  90  da.  at  3^. 

5.  $275.60  for  270  da.  at  4^^. 

6.  $385.75  for  200  da.  at  3^. 

7.  $75  for  2  yr.  210  da.  at  4^. 

8.  $85.50  for  3  yr.  20  da.  at  4^^. 

9.  $145  from  Sept.  15,  1902  to  Dec.  1,  1903,  at  5^. 

10.  $850.60  from  May  1,  1901  to  Sept.  1,  1903,  at  m. 

11.  $205.25  from  Jan.  2,  1902  to  March  1,  1904,  at  4^. 
SuG.     The  time  =  3  yr.  59  da.     Why  ? 

12.  $850.75  from  May  1,  1903  to  Dec.  31,  1903,  at  5^. 

BUSINESS  PAPERS. 
A  PROMISSOKY  NOTE. 


r^  cyu"iJLi-.&,iL<\.r<' )  — > — 


fiml?.) 


536.  A  Promissory  Note  is  a  written  promise  to  pay 
a  specified  sum  to  a  certain  party  at  a  certain  time. 


250  PERCENTAGE. 


637.  The  Maker  of  a  promissory  note  is  the  party 
who  signs  it  and  thereby  promises  to  pay.  The 
Payee  is  the  party  whose  name  is  written  in  the  body 
of  the  note  and  to  whom  the  promise  to  pay  is  made. 
The  Face  is  the  amount  named  in  the  note. 

Thus,  in  the  note  on  page  249  A.  P.  Reid  is  the  maker ,  Geo.  Morris 
Philips  the  j>ayee,  and  forty-five  r^  dollars  thd/ace. 

Note  1.  A  note  may  be  written  so  as  to  include  the  payment  of 
interest. 

Note  2.  If  a  note  is  payable  a  certain  number  of  days  after 
date,  it  is  due  at  the  expiration  of  that  number  of  days  after  date, 
not  counting  the  date. 

Thus,  a  note  payable  60  da.  after  June  1  is  due  July  31. 

Note  3.  If  a  note  is  payable  a  certain  number  of  months  after 
date,  calendar  months  are  to  be  counted. 

Thus,  a  note  payable  3  mo.  after  June  1  is  due  Aug.  1. 

A  CHECK. 


No.  /  -1  5  Lancaster,  Pa — £Luljjk_.aiv__190.a^ 

First  JknoNAL  Bi^KoflAif  caster 

Pay lO the Orderof  CjqrmJn  ^  l[f->VmA^nnnrn 


538.  A  Check  is  a  written  order  by  which  a  bank 
is  directed  by  a  depositor  to  pay  out  money  belong- 
ing to  him. 

639.  The  holder  of  a  check  should  have  it  cashed 


BUSINESS  PAPERS.  251 

on  the  day  he  receives  it,  or  on  the  first  business 
day  following. 

640.  Before  the  hold 


ia/KVUiiy  £^.   ci^eyTixLeyVa 


er  of  a  check  has  it 

cashed,  he  must  write 

his  name  across  the  back  of  it.      This  is  called 

indorsing  the  check. 

A  RECEIPT. 


541.  A  Receipt  is  a  written  acknowledgment  of 
money  or  goods  received,  or  of  services  rendered. 

Note.  When  the  money  paid  is  only  part  of  the  debt  owed,  "on 
account^''  should  be  written  in  the  receipt  instead  of  "  in  full  of  all 
accounts  to  date.''' 

542.  Exercise. 

/.  When  is  the  note  on  p.  249  due  ? 

2.  When  is  a  note  due  that  is  dated  July  1,  and  payable 
in  60  da.  ? 

3.  When  is  a  note  due  that  is  dated  May  1,  and  payable 
in  3  mo.  ? 


252  MISCELLANEOUS  PROBLEMS. 

4.  Who  pays  the  note  on  p.  249,  when  due  ? 

5.  Write  a  promissory  note  for  $25  with  yourself  as 
maker  and  L.  M.  Morris  as  payee,  dating  the  note  June  1, 
1903,  and  making  it  payable  in  90  da. 

6.  Suppose  you  buy  a  cow  from  F.  P.  Haws  for  $45.50, 
payable  in  G  mo. ;  write  the  promissory  note  that  you  could 
give  him  in  payment. 

7.  Suppose  you  have  money  deposited  in  the  Central 
National  Bank  of  Wilmington,  Del. ;  write  the  check  with 
which  you  could  pay  Joseph  S.  Walton  $10.  Show  how 
Joseph  S.  Walton  would  indorse  the  check  before  having 
it  cashed. 

8.  Write  a  receipt  showing  that  C.  V.  Miller  paid  you 
$75  to-day  on  account. 

9.  Write  a  receipt  for  $40  wages  that  were  due  you  for 
the  month  of  June,  1903,  from  L.  M.  Haines,  and  paid 
the  last  day  of  that  month. 


MISCELLANEOUS  PROBLEMS. 

543.  Oral  Exercise. 

/.  After  a  young  man's  salary  had  been  increased  ^  of 
itself,  it  was  $5.50  per  week  ;  what  was  it  before  the 
increase  ? 

2.  How  many  times  is  J  contained  in  1  ? 

3.  $1,000,000  is  how  many  times  $100,000  ? 

4.  If  an  automobile  runs  12  mi.  per  hour,  in  how  many 
minutes  will  it  run  a  mile  ? 


MISCELLANEOUS  PROBLEMS.  263 

5.  What  will  it  cost  to  send  by  mail  a  book  weighing 
19  oz.,  the  rate  of  postage  being  1  ct.  for  2  oz.  or  fraction 
thereof  ? 

6.  If  the  cost  of  sending  a  telegram  is  25  ct.  for  the  first 
10  words  and  2  ct.  for  each  additional  word  ;  find  the  cost 
of  sending  the  following  message  :  '^  Meet  me  at  my  office 
to-morrow  morning.     Bring  books.     Important  meeting. '* 

7.  What  should  1  doz.  loaves  of  bread  cost  when  3  loaves 
cost  10  ct.? 

8.  Supply  the  missing  numbers  and  read  : 

275  rails       = hundred  rails. 

87.5  pounds  = hundred  pounds. 

2735  shad       = hundred  shad. 

7567  plants    = thousand  plants. 

544.  Exercise. 

[From  United  States  Civil  Service  Examinations.] 

/.  Divide  2,408,588  by  4,732. 

2.  Multiply  8,643  by  608,  and  then  subtract  98,746. 

3.  A  merchant  who  spent  $225,  bought  65  pounds  of 
butter  at  30  cents  per  pound,  84  barrels  of  apples  at  12.25 
per  barrel,  and  spent  the  remainder  for  coffee.  How 
much  did  he  spend  for  coffee  ? 

4.  During  the  month  of  August  450,000  bushels  of  wheat 
were  shipped  from  a  certain  port.  During  September 
87,960  more  bushels  were  shipped  than  during  August. 
What  was  the  total  number  of  bushels  shipped  in  the  two 
months  ? 


254  MISCELLANEOUS  PROBLEMS. 

6.  Add  the  following,  placing  the  total  at  the  bottom  : 

742,155.74 

429.39 

6,873.68 

397.49 

1,956,374.20 

6.  If  a  railway  mail  clerk  earns  $800  in  a  year,  how  much 
will  he  have  left  after  paying  his  board  at  the  rate  of 
$16  a  month  ? 

7.  If  a  railway  mail  clerk  spends  ten  cents  a  day  for 
street-car  fare,  how  much  will  he  spend  in  six  months  of 
30  days  each  ? 

546..  Exercise. 

/.  If  4  men  can  do  a  piece  of  work  in  8J-  da.,  how  long 
should  it  take  1  man  to  do  it  ? 

2.  If  I  of  a  number  is  12 J,  what  is  f  of  the  number  ? 

3.  What  date  is  30  da.  prior  to  Dec.  31  ? 

4.  A  roll  of  wall  paper  is  8  yd.  long  and  18  in.  wide ; 
how  many  square  feet  does  it  contain  ? 

5.  AVhat  decimal  taken  100  times  will  make  .1  ? 

6.  How  many  2-inch  squares  can  be  cut  from  a  piece 
of  pasteboard  6  in.  long  and  5  in.  wide  ?  How  many 
square  inches  of  waste  will  there  be  ? 

7.  How  many  square  feet  of  canvas  is  required  to  cover 
a  box  6  ft.  6  in.  long,  4  ft.  wide,  and  3  ft.  deep  ? 

8.  What  must  be  added  to  .001  to  make  .1  ? 


MISCELLANEOUS  PROBLEMS.  255 

9.  A  man  who  takes  on  an  average  110  steps  per 
minute,  each  step  2J  ft,  long,  is  walking  at  the  rate  of 
how  many  miles  per  hour  ? 

10.  A  rectangular  piece  of  land  containing  1  A.  is  80  yd. 
long  ;  find  its  width  in  yards. 

//.  A  man  bought  a  piece  of  oil  cloth  containing  20  sq. 
yd. ;  what  is  its  length,  if  it  is  1 J  yd.  wide  ? 

546.  Exercise. 

/.  A  man  who  was  born  60  years  ago  has  lived  how 
many  hours,  supposing  that  there  have  been  14  leap  years 
in  the  60  years? 

2.  How  many  yards  long  is  a  tract  of  land  containing  an 
acre,  if  it  is  20  yards  wide? 

3.  2\  doz.  hats  cost  $72;  what  should  they  sell  for  apiece 
in  order  that  the  gain  may  be  $1  on  each  hat? 

4.  How  much  waste  will  there  be  after  cutting  as  many 
2-inch  cubes  as  possible  out  of  a  5-inch  cube? 

5.  A  common  brick  is  8  in.  long,  4  in.  wide,  and  2  in. 
thick;  how  many  square  inches  are  there  in  its  surface? 

6.  A  mechanic  was  employed  for  1  year  at  $60  a  month; 
if  his  expenses  were  $32  a  month,  how  much  did  he  save 
during  the  year? 

7.  Find  the  total  length  of  the  edges  of  a  box  6  in.  long, 
4  in.  wide,  and  3  in.  high. 

8.  12  bu.  of  wheat  were  raised  on  a  piece  of  land  121  yd. 
long  and  10  yd.  wide;  at  the  same  rate,  what  would  be  the 
yield  on  an  acre? 

9.  Show  by  an  arrangement  of  dots  that  3x2  =  2x3. 


256  MISCELLANEOUS  PROBLEMS. 

547.  Exercise. 

[From  United  States  Civil  Service  Examinations.] 
/.  Multiply  7^  by  36.8,  and  divide  the  product  by  1.92. 
(Solve  by  decimals.) 

2.  A  lot  which  was  53  feet  wide  and  150  ft.  long  sold 
for  $8347.50,  which  was  one-fourth  more  than  it  cost. 
What  was  the  cost  per  square  foot  ? 

3.  Multiply  382.58  by  f  of  27,342,  and  divide  the  prod- 
uct by  J  of  34.78.     (Work  by  decimals.) 

4.  Divide  5f  by  ^,  multiply  the  quotient  by  3.5468,  and 
from  the  product  subtract  \^  of  13.70. 

5.  Multiply  37f  by  400.3,  and  divide  the  product  by 
93.5.     (Solve  by  decimals.) 

6.  A  carrier  can  assort  43  letters  or  37  papers  in  a 
minute.  At  this  rate,  how  many  hours  will  it  take  him 
to  assort  3655  letters  and  185  pounds  of  papers,  averaging 
7  papers  to  the  pound  ? 

7.  In  a  certain  mail  there  are  294  pounds  14  ounces  of 
newspapers  weighing  at  the  rate  of  3  papers  to  every  7 
ounces.  How  many  papers  are  there  in  the  mail  ?  (16 
ounces  =  1  pound.) 

8.  If  a  cubic  foot  of  coal  weighs  63  pounds,  find  the 
number  of  tons  of  coal  in  a  bin  5  feet  6  inches  wide,  6  feet 
9  inches  deep,  and  19  feet  6  inches  long.  (1  ton  =  2,240 
pounds.) 


VB  35879 


^*i<^"f;.-;^ii'-«'' 


5ll;H3 


UNIVERSITY  OF  CALIFORNIA  UBRARY 


